Medicare overview

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Language: en

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SALMAN KHAN: I'm here with Dr.
Laurence Baker from Stanford

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Medical School, and we're
going to talk about what?

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DR. LAURENCE BAKER:
How about Medicare?

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SALMAN KHAN: Medicare.

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I think we knew that was
going to happen, because we

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had this picture here already.

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So what is this a picture of?

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DR. LAURENCE BAKER: I think
that's a picture of LBJ

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signing the Medicare
Act in 1965.

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This is the time that
we passed Medicare.

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SALMAN KHAN: Lyndon
Baines Johnson, LBJ.

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This is Truman right over here.

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DR. LAURENCE BAKER: I think so.

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SALMAN KHAN: Maybe he was there
because I think he was probably

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old enough by that
point to be a recipient.

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DR. LAURENCE BAKER: Yeah.

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65 and over.

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SALMAN KHAN: So, I
think we all have

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a general idea of what Medicare
is, that it's kind of a health

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care program, or insurance
program for people

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who are retired.

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But then, it starts to
get fuzzy after that.

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There are all these details
that confuse most of us.

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DR. LAURENCE BAKER: Yeah,
so we talked about Medicare

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in a bunch of different ways.

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Medicare is a program.

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It's a US government program
run by the federal government.

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SALMAN KHAN: So it is run
by the federal government?

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DR. LAURENCE BAKER: Yep.

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SALMAN KHAN: And sometimes,
it's thrown in with Medicaid.

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How are they different,
just at a very high level.

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DR. LAURENCE BAKER:
Oh, so Medicaid

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is another different
insurance program run

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by the federal government
in coordination

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with state governments.

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Medicaid is a federal
and state program.

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Medicare is a federal program.

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Medicare is for folks who
are 65 years old and older

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and people with disabilities.

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Principally those two groups.

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Medicaid is a program
that's aimed at lower income

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populations, and so--

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SALMAN KHAN: OK.

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So this older people
and disabilities.

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This is low income,
kind of a poverty issue.

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DR. LAURENCE BAKER: Yes.

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Mostly.

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SALMAN KHAN: I'm
gonna write that down.

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Low income.

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And this is primarily retirees?

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DR. LAURENCE BAKER: Yeah.

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The vast majority
of people in there

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are folks who are there
because they are over 65.

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SALMAN KHAN: OK.

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And we're just going
to focus on Medicare.

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And Medicare is
federal government.

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Medicaid is the federal
and the state government

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are on Medicaid, because they
always sound very similar.

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DR. LAURENCE BAKER:
They sound very similar.

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Yeah, I know.

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They're easily confused,
certainly at the name level.

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Really different programs.

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They operate
completely separately.

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In a few cases, there are people
who get access to both of them,

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but they pay differently.

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They run differently.

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They're--

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SALMAN KHAN: But
at federal level,

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they are still run by
the same organization,

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the Centers for
Medicaid and Medicare.

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Or Medicaid--

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DR. LAURENCE BAKER: Yeah.

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The Department of Health
and Human Services

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has a subpart called the Centers
for Medicare and Medicaid

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Services.

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SALMAN KHAN: --executive branch.

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DR. LAURENCE BAKER: Yeah.

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SALMAN KHAN: The
executive branch

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of our government, obviously.

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And then you have the
Secretary of Health--

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DR. LAURENCE BAKER:
Health and Human Services.

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SALMAN KHAN: So, Health
and Human Services.

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And then within
that, you have CMS.

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DR. LAURENCE BAKER:
You have CMS.

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SALMAN KHAN: Which
is the Centers for--

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and you were saying earlier,
it should have two M's in it.

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DR. LAURENCE BAKER: Yeah.

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It's the Centers for Medicare
and Medicaid Services.

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So I think it's just
shorter to have one M.

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But really, there's
two in there.

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SALMAN KHAN: [INAUDIBLE]
instead of calling it CMMS.

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OK.

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They administer at the
federal level, both Medicare

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and Medicaid.

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But just to reiterate,
Medicare is really

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mainly a federal thing,
while Medicaid is both.

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DR. LAURENCE BAKER:
Yeah, Medicare--

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SALMAN KHAN: And
they're run separately,

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so you have Medicare here.

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Medicare.

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And then you Medicaid here.

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DR. LAURENCE BAKER: Yeah.

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SALMAN KHAN: I see.

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All right.

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All right.

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So let's dive deep
into Medicare.

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So I guess, what is it?

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DR. LAURENCE BAKER: So Medicare
is an insurance program.

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It provides insurance
coverage for people

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who are eligible for it.

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So if you're age 67
and over, or if you

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have a permanent disability,
a couple other small groups.

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If you have Lou Gehrig's
disease, or end stage

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renal disease, kidney
failure, then you

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can get access to
Medicare too, but those

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are small populations.

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So if you're in those groups,
you're eligible for Medicare.

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Medicare will provide
you with insurance

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that will cover hospital bills,
doctor bills, some prescription

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drugs.

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It'll cover lots
of medical care--

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SALMAN KHAN: You know,
I hear these things--

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this is the part that I think
gets confusing for everyone.

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Is that you start hearing Part
A, Part B, Part C, Part, D.

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I don't even if those
are all the parts.

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What are those referring to?

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And are they exactly
insurance, or are they

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something slightly different?

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DR. LAURENCE BAKER: OK, so
Medicare has these four parts.

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Part A, Part B,
Part C, and Part D.

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And it can get a
little confusing,

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but we can break it up
into some different pieces.

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So the biggest one, in terms
of the number of people

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who are in it, is what sometimes
we call traditional Medicare.

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And that's what historically
is Part A and Part B.

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And you kind of want to
take those two together,

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because they make a package.

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And we call that maybe
traditional Medicare.

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Sometimes people
call it original

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Medicare, standard Medicare.

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But when they
passed the program,

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when they first
did it, Part A was

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meant to cover
hospital expenditures,

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and Part B was aimed
at doctor costs, costs

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for seeing physicians.

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Of course, those things
go so closely together

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that nowadays, Part A and
Part B are kind of a pair.

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So you really rarely would find
someone who only wanted Part A,

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or only wanted Part B.

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SALMAN KHAN: Right.

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Sometimes, you hear the doctor
processes it as outpatient.

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DR. LAURENCE BAKER:
It's [INAUDIBLE].

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SALMAN KHAN: Hospitals,
sometimes you

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hear the word inpatient,
means you're living there

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while they're treating you.

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Outpatient, you go visit them
for an hour or two and leave.

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And that's where you hear those.

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DR. LAURENCE BAKER: Yeah.

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So that's a way of breaking
up the kinds of services

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that medicine provides.

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Medicare actually gets a little
bit-- it's mix and match.

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So really, Medicare
Part A expenditures

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when a hospital bills Medicare
for use of the hospital, use

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of a room, use of the
services at the hospital.

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Anytime a doctor bills Medicare,
even if they saw the patient

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in the hospital, they're going
to be a Part B expenditure.

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SALMAN KHAN: I see.

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I see.

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So it isn't strictly
inpatient, outpatient.

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It really is much more along the
lines of hospital and doctor.

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DR. LAURENCE BAKER: Yep.

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SALMAN KHAN: OK.

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So I'll maybe put this in
some type of [INAUDIBLE] way

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or something.

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It's not necessarily
outpatient, inpatient.

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That's maybe one way of--

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DR. LAURENCE BAKER: Yes,
it is useful to think

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about that,
outpatient, inpatient.

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SALMAN KHAN: And will it
cover everything that happens?

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Including drugs?

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DR. LAURENCE BAKER: OK.

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So drugs is one key issue.

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To the main question, the
first thing I guess I'd say

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is that traditional Medicare
is pretty comprehensive.

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It covers most of the things
that people would normally

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encounter in their needs
throughout the year.

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SALMAN KHAN: Right.

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Does it cover everything?

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DR. LAURENCE BAKER: Prescription
drugs is one big exception.

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SALMAN KHAN: It does not
cover-- so Parts A and Part

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B do not cover
prescription drugs.

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So when Lyndon Johnson
was signing this,

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if Truman had to get
prescribed some drug,

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and if he was a Medicare
recipient, in 1965,

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he would have had to
pay that out of pocket.

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DR. LAURENCE BAKER: Yeah.

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So I guess one
caveat I'll note is

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that if you get prescription
drugs that are administered

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to you while you're
in the hospital--

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SALMAN KHAN: I see.

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DR. LAURENCE BAKER: --you're
staying in the hospital,

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they give you drugs, that's
going to be generally covered

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under Part A. It's
when you're outpatient,

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and you're going home, and
somebody sends you to pharmacy

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to pick up a--

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SALMAN KHAN: I see.

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Medicare in its original
form would not cover that.

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DR. LAURENCE BAKER:
They would not

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cover the outpatient
prescription drugs.

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So that became a
problem for some folks,

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and that has generated changes
to the Medicare program.

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SALMAN KHAN: Right.

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Since then, there is a
part now that cover that.

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DR. LAURENCE BAKER: Yes.

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So that's the Part D program
as a Medicare extension.

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I don't know how far
you want to get--

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SALMAN KHAN: We can-- I'll
write that right here.

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Well just, D for drugs.

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DR. LAURENCE BAKER: Yeah.

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The whole story of
prescription drugs and Medicare

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has a lot to it.

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People would buy
additional policies

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if they had A and B.
They'd buy themselves

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a private supplement that
might cover drugs sometimes.

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But now, with the
new Part D plan,

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maybe people would
buy that instead.

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And there's other stuff.

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SALMAN KHAN: There's
probably a C right there.

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DR. LAURENCE BAKER:
There's a C that's coming.

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SALMAN KHAN: We can get
to that in a second.

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But so, A and B, traditional.

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A, hospitals.

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B, doctors.

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And what percent would--
with my health insurance,

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there's a copay every
time I visit the a doctor.

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It's $20.

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How does it work with
traditional Medicare,

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with Parts A and Part B?

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DR. LAURENCE BAKER: So if you
just have Part A and Part B,

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then on the Part B
side, doctors are

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covered 80% of the
Medicare allowable charges.

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So Medicare has a
fee schedule that

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governs what doctors can
be paid for services.

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And Medicare will cover 80% of
that, and the patient pays 20%.

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SALMAN KHAN: I see.

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DR. LAURENCE BAKER:
For Part A, it

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gets a little more complicated,
but there's a co-payment

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that you have to make.

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If you go to the
hospital, Medicare

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is going to have a
copayment of a fixed

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amount, several hundred
dollars, coming up.

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SALMAN KHAN: OK.

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It's just not as
simple as 80-20.

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DR. LAURENCE BAKER: No.

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Not as simple as 80-20.

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You're going to pay a
copayment there too.

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So they're not bad coverage.

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But there's certainly
some costs you can run up.

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And if you're pretty
sick, you might really

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have a bill that's
left [INAUDIBLE].

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SALMAN KHAN: And there's
extreme flexibility here.

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You can see any doctor you want,
go to any hospital you want.

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DR. LAURENCE BAKER: Yeah.

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Medicare Part A and Part B
is, to one way of thinking,

00:08:08.650 --> 00:08:11.190
one of the last bastions of
health care as it exists,

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or health insurance as it
existed in America in 1965,

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'70, '75, when there were really
minimal restrictions on what

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doctors you could see, what
hospitals you could use.

00:08:20.220 --> 00:08:23.541
Doctors get lots of discretion
here about the services

00:08:23.541 --> 00:08:26.040
they're going to provide, and
there's not a lot of oversight

00:08:26.040 --> 00:08:28.400
that you might see in today's
HMOs, or other kinds of--

00:08:28.400 --> 00:08:28.570
SALMAN KHAN: Right.

00:08:28.570 --> 00:08:30.180
But there are still--
maybe I'm wrong here--

00:08:30.180 --> 00:08:32.570
there still are doctors that
won't take Medicare patients?

00:08:32.570 --> 00:08:34.028
DR. LAURENCE BAKER:
So, doctors can

00:08:34.028 --> 00:08:36.700
elect to be Medicare
doctors or not.

00:08:36.700 --> 00:08:38.387
And some doctors,
I think, decide

00:08:38.387 --> 00:08:40.220
they have enough business
or enough patients

00:08:40.220 --> 00:08:40.969
from other places.

00:08:40.969 --> 00:08:42.915
And Medicare's a plan
that's got its rules,

00:08:42.915 --> 00:08:44.290
and you have to
follow the rules.

00:08:44.290 --> 00:08:45.892
And some people
think that Medicare

00:08:45.892 --> 00:08:47.850
doesn't pay as well as
other options they have.

00:08:47.850 --> 00:08:49.107
So there are a few.

00:08:49.107 --> 00:08:50.690
Most of the doctors,
the vast majority

00:08:50.690 --> 00:08:52.470
of doctors in the country,
will work with Medicare.

00:08:52.470 --> 00:08:53.550
SALMAN KHAN: And
you're saying, it

00:08:53.550 --> 00:08:55.060
is kind of the super
flexible thing.

00:08:55.060 --> 00:08:58.727
If you do have an illness, you
can go see multiple doctors,

00:08:58.727 --> 00:08:59.560
get second opinions.

00:08:59.560 --> 00:09:02.560
You could go to the specialist
on the other side the country.

00:09:02.560 --> 00:09:04.420
So it is pretty good that way.

00:09:04.420 --> 00:09:07.930
Although you have to pay 20%
of at least the doctor's visit,

00:09:07.930 --> 00:09:13.110
so it's a pretty-- so what
happens if I'm a retiree?

00:09:13.110 --> 00:09:14.650
I don't have a lot of money.

00:09:14.650 --> 00:09:15.520
I have a major bill.

00:09:15.520 --> 00:09:18.260
Say I have a surgery
that's $50,000.

00:09:18.260 --> 00:09:22.120
I know that's more of a
hospital inpatient thing.

00:09:22.120 --> 00:09:23.800
Are there other
options so that they

00:09:23.800 --> 00:09:26.719
don't have to pay
that out of pocket?

00:09:26.719 --> 00:09:27.760
DR. LAURENCE BAKER: Yeah.

00:09:27.760 --> 00:09:28.660
So there's a bunch
of things that

00:09:28.660 --> 00:09:31.076
go into trying to figure out
what the cost to getting this

00:09:31.076 --> 00:09:33.320
are, and how you
might get covered

00:09:33.320 --> 00:09:35.029
for some of these
other services.

00:09:35.029 --> 00:09:37.320
The main thing that would be
the answer to the question

00:09:37.320 --> 00:09:39.900
is, there are
supplemental policies.

00:09:39.900 --> 00:09:41.870
So if you're a
retiree, and you've

00:09:41.870 --> 00:09:44.642
gotten Part A, because
you're eligible for that,

00:09:44.642 --> 00:09:46.100
and you've Medicare
tax, and you've

00:09:46.100 --> 00:09:47.730
got Part B, for
which you do have

00:09:47.730 --> 00:09:50.842
to pay a little bit of money
every month-- subsidized,

00:09:50.842 --> 00:09:52.550
but still a premium
you have to pay-- you

00:09:52.550 --> 00:09:54.840
might decide in
addition to that to pay

00:09:54.840 --> 00:09:57.210
for a supplemental
insurance policy, which

00:09:57.210 --> 00:09:59.760
you'd buy from a private place.

00:09:59.760 --> 00:10:01.959
The AARP is one place
that sells these.

00:10:01.959 --> 00:10:03.500
There are other
folks that sell them.

00:10:03.500 --> 00:10:05.100
You pay an extra
premium for that.

00:10:05.100 --> 00:10:07.820
That's an additional
insurance plan that you have.

00:10:07.820 --> 00:10:11.270
And that might agree, or you
might have a method there

00:10:11.270 --> 00:10:13.530
to have them cover
that 20% co-pay,

00:10:13.530 --> 00:10:14.760
the additional hospitals.

00:10:14.760 --> 00:10:16.370
And those plans, there's
some different varieties

00:10:16.370 --> 00:10:18.610
of those that cover more or
less generously, and of course,

00:10:18.610 --> 00:10:20.249
charge a higher or
a lower premium for.

00:10:20.249 --> 00:10:21.040
SALMAN KHAN: I see.

00:10:21.040 --> 00:10:22.360
That makes complete sense.

00:10:22.360 --> 00:10:26.049
And do people have to pay any
premium to be in Medicare?

00:10:26.049 --> 00:10:28.340
DR. LAURENCE BAKER: To get
A and B both, you have to A,

00:10:28.340 --> 00:10:29.880
you generally don't.

00:10:29.880 --> 00:10:33.589
If you pay your Medicare tax for
10 years, you work for 10 years

00:10:33.589 --> 00:10:36.130
here, and you pay your Medicare
tax, you get Part A for free.

00:10:36.130 --> 00:10:38.870
But Part B charges a premium.

00:10:38.870 --> 00:10:40.820
So you do have to
pay a small monthly--

00:10:40.820 --> 00:10:41.440
SALMAN KHAN: And it's what?

00:10:41.440 --> 00:10:43.100
It's like on the order
of like, $100 a month?

00:10:43.100 --> 00:10:43.830
DR. LAURENCE BAKER:
Yeah, generally.

00:10:43.830 --> 00:10:46.440
It's gone up a little bit over
time, or something like that.

00:10:46.440 --> 00:10:47.856
SALMAN KHAN: So
it's not too much.

00:10:47.856 --> 00:10:51.020
Maybe $100 or a little bit more
per month, which is nothing.

00:10:51.020 --> 00:10:53.482
I mean, if you're
65 or over, and you

00:10:53.482 --> 00:10:54.940
try to get a private
policy, you're

00:10:54.940 --> 00:10:56.610
probably looking at something
like a thousand dollars

00:10:56.610 --> 00:10:57.060
a month.

00:10:57.060 --> 00:10:57.300
DR. LAURENCE BAKER: Yes.

00:10:57.300 --> 00:10:59.170
So I guess two ways
of thinking about it.

00:10:59.170 --> 00:11:00.503
One, this is heavily subsidized.

00:11:00.503 --> 00:11:03.230
Compared to what a
person 75 years old or 80

00:11:03.230 --> 00:11:04.970
would pay for an actual
private insurance

00:11:04.970 --> 00:11:07.127
policy would be much,
much higher than that.

00:11:07.127 --> 00:11:08.710
There are people who
look at that $100

00:11:08.710 --> 00:11:10.950
a month in fixed income and
Social Security and things,

00:11:10.950 --> 00:11:12.116
and you say that's an issue.

00:11:12.116 --> 00:11:14.202
And so one of the things
that's just in the news

00:11:14.202 --> 00:11:16.660
the last few weeks is the fact
that this number, the number

00:11:16.660 --> 00:11:17.824
went up a bit.

00:11:17.824 --> 00:11:19.740
Social Security payments
went up a little bit.

00:11:19.740 --> 00:11:21.010
But less, actually,
than the Medicare.

00:11:21.010 --> 00:11:21.360
So people are saying--

00:11:21.360 --> 00:11:21.710
SALMAN KHAN: I see.

00:11:21.710 --> 00:11:22.490
So there's a net decrease.

00:11:22.490 --> 00:11:22.820
I see.

00:11:22.820 --> 00:11:23.050
I see.

00:11:23.050 --> 00:11:24.883
DR. LAURENCE BAKER:
--if you're buying both.

00:11:24.883 --> 00:11:26.530
So there's some
people who will still

00:11:26.530 --> 00:11:27.850
be worried about
that, even though it

00:11:27.850 --> 00:11:30.020
is heavily subsidized, and
looks like a great deal.

00:11:30.020 --> 00:11:30.811
SALMAN KHAN: I see.

00:11:30.811 --> 00:11:32.712
And then finally,
so we have no A, B.

00:11:32.712 --> 00:11:34.170
And then we have
a little bit of D.

00:11:34.170 --> 00:11:36.130
We know it's all
about covering drugs.

00:11:36.130 --> 00:11:37.260
What does C do?

00:11:37.260 --> 00:11:38.011
[INAUDIBLE] exist?

00:11:38.011 --> 00:11:39.010
DR. LAURENCE BAKER: Yes.

00:11:39.010 --> 00:11:39.785
C exists.

00:11:39.785 --> 00:11:42.710
C has been around for
20 or 30 years now.

00:11:42.710 --> 00:11:44.470
C is a separate choice.

00:11:44.470 --> 00:11:46.120
So when you turn
65, let's say, you

00:11:46.120 --> 00:11:48.620
can make this choice a little
bit differently sometimes too.

00:11:48.620 --> 00:11:50.720
But when you turn
65, you might decide

00:11:50.720 --> 00:11:53.270
to take part A and part B, maybe
by a traditional supplement.

00:11:53.270 --> 00:11:55.155
Maybe you add the first
Part D drug plan on

00:11:55.155 --> 00:11:56.238
and make yourself package.

00:11:56.238 --> 00:12:00.460
Or you can elect Part C, and
Part C is an alternative.

00:12:00.460 --> 00:12:04.220
Under Part C, what happens is
the CMS, federal government,

00:12:04.220 --> 00:12:06.250
has contracted with some HMOs.

00:12:06.250 --> 00:12:08.394
Kaiser, Blue Cross,
other places.

00:12:08.394 --> 00:12:10.310
Sometimes, they've
contracted with a few PPOs.

00:12:10.310 --> 00:12:12.000
They've tried other
kinds of plans too.

00:12:12.000 --> 00:12:13.580
Mostly it's HMOs.

00:12:13.580 --> 00:12:15.170
So you have a choice then.

00:12:15.170 --> 00:12:21.137
You can join Kaiser, and have
Kaiser be your Medicare plan.

00:12:21.137 --> 00:12:22.470
You can join the Blue Cross one.

00:12:22.470 --> 00:12:23.600
Depending on where
you are, you'll

00:12:23.600 --> 00:12:25.641
have different choices of
different private plans

00:12:25.641 --> 00:12:26.760
that have contracted.

00:12:26.760 --> 00:12:27.830
And you can do that.

00:12:27.830 --> 00:12:30.450
The federal government will
then pay a bunch of your premium

00:12:30.450 --> 00:12:31.290
to them.

00:12:31.290 --> 00:12:31.650
SALMAN KHAN: I see.

00:12:31.650 --> 00:12:34.270
DR. LAURENCE BAKER: So you'll
get covered by the private plan

00:12:34.270 --> 00:12:36.269
with the federal government
covering your costs.

00:12:36.269 --> 00:12:38.490
They might ask you to pay
some additional premium.

00:12:38.490 --> 00:12:40.948
You're going to have to pay
your Part B equivalent already.

00:12:40.948 --> 00:12:42.550
So you've got to
pay that amount.

00:12:42.550 --> 00:12:44.710
But then, you might have
to pay a little bit extra.

00:12:44.710 --> 00:12:47.156
And then, you'll get access
to what services that has.

00:12:47.156 --> 00:12:48.780
And for some people,
that's attractive.

00:12:48.780 --> 00:12:52.190
Because though the co-payments
and deductibles will be lower--

00:12:52.190 --> 00:12:54.932
you'll often have more
like the $10 copay--

00:12:54.932 --> 00:12:56.890
you might have access to
some prescription drug

00:12:56.890 --> 00:12:58.920
coverage automatically
built in there.

00:12:58.920 --> 00:13:01.237
Sometimes those cover some
vision and other things,

00:13:01.237 --> 00:13:02.070
which might be nice.

00:13:02.070 --> 00:13:04.370
SALMAN KHAN: So
this one or plans.

00:13:04.370 --> 00:13:07.160
I could get this, while
it's 80% coverage-- well,

00:13:07.160 --> 00:13:09.160
I can't make that rule,
because hospitals, it

00:13:09.160 --> 00:13:10.110
could be different.

00:13:10.110 --> 00:13:11.776
But you get this kind
of flexible thing.

00:13:11.776 --> 00:13:13.720
But you have a pretty
significant co-pay,

00:13:13.720 --> 00:13:14.790
the 20% patient.

00:13:14.790 --> 00:13:16.830
I could get supplemental
insurance for me

00:13:16.830 --> 00:13:18.960
to cover some of that
right over there.

00:13:18.960 --> 00:13:21.169
Or I can do Part C. Is
there a name for Part C?

00:13:21.169 --> 00:13:22.710
DR. LAURENCE BAKER:
Part C these days

00:13:22.710 --> 00:13:24.170
is called Medicare Advantage.

00:13:24.170 --> 00:13:25.890
So the plans in
Part C are called

00:13:25.890 --> 00:13:27.780
Medicare Advantage plans.

00:13:27.780 --> 00:13:30.790
SALMAN KHAN: Which is
really just Medicare

00:13:30.790 --> 00:13:32.977
subsidizing me to get
private insurance.

00:13:32.977 --> 00:13:34.060
DR. LAURENCE BAKER: Right.

00:13:34.060 --> 00:13:36.960
From some plans that have agreed
to work with Medicare patients.

00:13:36.960 --> 00:13:40.380
SALMAN KHAN: And there's
various checking up on them.

00:13:40.380 --> 00:13:42.270
So they agree to
basically cover everything

00:13:42.270 --> 00:13:44.080
that A and B would
cover, so you're not

00:13:44.080 --> 00:13:45.680
going to get
skimped on too much.

00:13:45.680 --> 00:13:46.300
SALMAN KHAN: And what
do most people do?

00:13:46.300 --> 00:13:47.120
Do they go A and B?

00:13:47.120 --> 00:13:47.760
The traditional route?

00:13:47.760 --> 00:13:48.780
Or do they go the C route?

00:13:48.780 --> 00:13:50.610
DR. LAURENCE BAKER: So
historically, the vast majority

00:13:50.610 --> 00:13:52.250
of people have
gone A and B route.

00:13:52.250 --> 00:13:52.760
It changes.

00:13:52.760 --> 00:13:54.030
It kind of goes up
and down over time,

00:13:54.030 --> 00:13:55.310
depending on various factors.

00:13:55.310 --> 00:13:58.066
But over a historical
average, it

00:13:58.066 --> 00:14:02.740
would be something like 80%,
85%, 90% in A, and maybe 20%,

00:14:02.740 --> 00:14:05.470
15%, 10% in B and C,
depending on which year

00:14:05.470 --> 00:14:07.540
you're looking at
over the last 15 year.

00:14:07.540 --> 00:14:07.710
SALMAN KHAN: Right.

00:14:07.710 --> 00:14:10.043
And we could probably do a
whole series of videos on it.

00:14:10.043 --> 00:14:11.430
So when did D come about?

00:14:11.430 --> 00:14:13.180
It wasn't part of the original--

00:14:13.180 --> 00:14:15.305
DR. LAURENCE BAKER: D was
not part of the original.

00:14:15.305 --> 00:14:18.560
D comes about in the early
2000s under George Bush.

00:14:18.560 --> 00:14:19.982
George W. Bush.

00:14:19.982 --> 00:14:21.690
SALMAN KHAN: This is
what we heard about.

00:14:21.690 --> 00:14:22.740
I think it was like, 2003.

00:14:22.740 --> 00:14:23.410
I think I it was--

00:14:23.410 --> 00:14:25.743
DR. LAURENCE BAKER: I think
that's probably about right.

00:14:25.743 --> 00:14:27.750
And the issue there was
there's a lot of folks

00:14:27.750 --> 00:14:31.320
who were having trouble getting
their outpatient drugs covered.

00:14:31.320 --> 00:14:33.980
So this is a plan, this is
another part of Medicare.

00:14:33.980 --> 00:14:35.140
You can elect to buy this.

00:14:35.140 --> 00:14:38.030
You normally would think
about this if you got A and B,

00:14:38.030 --> 00:14:40.030
and you wanted to get
outpatient drugs covered,

00:14:40.030 --> 00:14:41.360
you'd think about a D plan.

00:14:41.360 --> 00:14:44.410
Part D is run by the federal
government, kind of the way

00:14:44.410 --> 00:14:45.370
that C is.

00:14:45.370 --> 00:14:48.790
They subsidize, and
they pay a premium

00:14:48.790 --> 00:14:53.149
to the private plans
that cover drugs.

00:14:53.149 --> 00:14:54.690
Or at least they
organize the market,

00:14:54.690 --> 00:14:56.980
and the person then pays
also some premium to get--

00:14:56.980 --> 00:14:57.170
SALMAN KHAN: I see.

00:14:57.170 --> 00:14:57.870
So a private--

00:14:57.870 --> 00:14:58.740
DR. LAURENCE BAKER:
So it's private.

00:14:58.740 --> 00:14:59.550
SALMAN KHAN: A
private entity will

00:14:59.550 --> 00:15:01.720
handle the actual
reimbursements for the drugs,

00:15:01.720 --> 00:15:03.370
but Medicare will pay them.

00:15:03.370 --> 00:15:03.650
DR. LAURENCE BAKER: Yeah.

00:15:03.650 --> 00:15:05.270
So the way to think
about it is, just

00:15:05.270 --> 00:15:06.860
a bunch of private
entities with funds

00:15:06.860 --> 00:15:10.500
flows coming in, partly from the
person, partly from Medicare,

00:15:10.500 --> 00:15:11.300
to buy these.

00:15:11.300 --> 00:15:12.280
SALMAN KHAN: OK.

00:15:12.280 --> 00:15:15.002
And who do you
pay your money to?

00:15:15.002 --> 00:15:15.960
You pay it to Medicare?

00:15:15.960 --> 00:15:16.834
You pay it to these--

00:15:19.250 --> 00:15:21.480
DR. LAURENCE BAKER: Uh.

00:15:21.480 --> 00:15:23.630
SALMAN KHAN:
Actually-- it works out

00:15:23.630 --> 00:15:26.570
the same way one
way or the other.

00:15:26.570 --> 00:15:27.820
This is optional.

00:15:27.820 --> 00:15:30.220
A retiree or someone
over 65 would pay money

00:15:30.220 --> 00:15:33.070
to essentially get a
drug plan subsidized.

00:15:33.070 --> 00:15:34.070
DR. LAURENCE BAKER: Yes.

00:15:34.070 --> 00:15:34.640
SALMAN KHAN: OK.

00:15:34.640 --> 00:15:35.160
Cool.

00:15:35.160 --> 00:15:35.660
Cool.

00:15:35.660 --> 00:15:36.720
Well, no, this is super.

00:15:36.720 --> 00:15:37.095
DR. LAURENCE BAKER:
Well, actually, we're

00:15:37.095 --> 00:15:37.830
trying to think about subsidies.

00:15:37.830 --> 00:15:39.380
But there's some interesting--

00:15:39.380 --> 00:15:40.270
[INTERPOSING VOICES]

00:15:40.270 --> 00:15:42.800
SALMAN KHAN: Oh, I guess to be
able to participate in a drug

00:15:42.800 --> 00:15:43.220
plan.

00:15:43.220 --> 00:15:44.120
DR. LAURENCE BAKER: I
mean, they are subsidized.

00:15:44.120 --> 00:15:46.460
And the federal is taking
a bunch of money in.

00:15:46.460 --> 00:15:49.215
But how exactly the economics
of these plans works out

00:15:49.215 --> 00:15:50.946
has been an interesting
discussion over the last couple

00:15:50.946 --> 00:15:51.020
over year.

00:15:51.020 --> 00:15:51.110
SALMAN KHAN: Right.

00:15:51.110 --> 00:15:51.230
Right.

00:15:51.230 --> 00:15:51.890
No, we could make
a ton of videos.

00:15:51.890 --> 00:15:52.790
No, this is super helpful.

00:15:52.790 --> 00:15:54.248
Because, I think,
frankly, I didn't

00:15:54.248 --> 00:15:57.570
know about the differences right
here before about 15 minutes

00:15:57.570 --> 00:15:58.441
ago.

00:15:58.441 --> 00:15:58.940
All right.

00:15:58.940 --> 00:16:00.990
Well, thanks a bunch.


Proof: Opposite angles of parallelogram congruent

WEBVTT
Kind: captions
Language: en

00:00:00.204 --> 00:00:01.620
What I want to do
in this video is

00:00:01.620 --> 00:00:04.270
prove that the opposite
angles of a parallelogram

00:00:04.270 --> 00:00:05.120
are congruent.

00:00:05.120 --> 00:00:10.790
So for example, we want to prove
that CAB is congruent to BDC,

00:00:10.790 --> 00:00:12.940
so that that angle is
equal to that angle,

00:00:12.940 --> 00:00:15.100
and that ABD, which
is this angle,

00:00:15.100 --> 00:00:18.470
is congruent to DCA, which
is this angle over here.

00:00:18.470 --> 00:00:20.160
And to do that, we
just have to realize

00:00:20.160 --> 00:00:21.910
that we have some
parallel lines,

00:00:21.910 --> 00:00:23.220
and we have some transversals.

00:00:23.220 --> 00:00:25.090
And the parallel lines
and the transversals

00:00:25.090 --> 00:00:26.024
actually switch roles.

00:00:26.024 --> 00:00:28.440
So let's just continue these
so it looks a little bit more

00:00:28.440 --> 00:00:33.461
like transversals
intersecting parallel lines.

00:00:33.461 --> 00:00:35.460
And really, you could
just pause it for yourself

00:00:35.460 --> 00:00:37.930
and try to prove it, because
it really just comes out

00:00:37.930 --> 00:00:40.670
of alternate interior angles
and corresponding angles

00:00:40.670 --> 00:00:43.740
of transversals
intersecting parallel lines.

00:00:43.740 --> 00:00:47.184
So let's say that this
angle right over here--

00:00:47.184 --> 00:00:49.100
let me do it in a new
color since I've already

00:00:49.100 --> 00:00:50.610
used that yellow.

00:00:50.610 --> 00:00:53.247
So let's start right
here with angle BDC.

00:00:56.270 --> 00:00:58.740
And I'm just going
to mark this up here.

00:00:58.740 --> 00:01:03.770
Angle BDC, right over here-- it
is an alternate interior angle

00:01:03.770 --> 00:01:07.050
with this angle right over here.

00:01:07.050 --> 00:01:09.550
And actually, we could
extend this point over here.

00:01:09.550 --> 00:01:13.920
I could call this
point E, if I want.

00:01:13.920 --> 00:01:20.710
So I could say angle CDB
is congruent to angle

00:01:20.710 --> 00:01:30.650
EBD by alternate
interior angles.

00:01:30.650 --> 00:01:31.720
This is a transversal.

00:01:31.720 --> 00:01:33.020
These two lines are parallel.

00:01:33.020 --> 00:01:36.680
AB or AE is parallel to CD.

00:01:36.680 --> 00:01:37.660
Fair enough.

00:01:37.660 --> 00:01:40.300
Now, if we kind of change
our thinking a little bit

00:01:40.300 --> 00:01:45.380
and instead, we now view BD
and AC as the parallel lines

00:01:45.380 --> 00:01:49.520
and now view AB as
the transversal,

00:01:49.520 --> 00:01:52.930
then we see that
angle EBD is going

00:01:52.930 --> 00:01:56.170
to be congruent to
angle BAC, because they

00:01:56.170 --> 00:01:57.435
are corresponding angles.

00:01:59.980 --> 00:02:10.560
So angle EBD is going to
be congruent to angle BAC,

00:02:10.560 --> 00:02:11.520
or I could say CAB.

00:02:14.680 --> 00:02:18.012
They are corresponding angles.

00:02:18.012 --> 00:02:19.970
And so if this angle is
congruent to that angle

00:02:19.970 --> 00:02:21.230
and that angle is
congruent to that angle,

00:02:21.230 --> 00:02:23.110
then they are congruent
to each other.

00:02:23.110 --> 00:02:30.380
So angle-- let me make sure
I get this right-- CDB,

00:02:30.380 --> 00:02:36.540
or we could say BDC, is
congruent to angle CAB.

00:02:36.540 --> 00:02:39.012
So we've proven this first
part right over here.

00:02:39.012 --> 00:02:40.970
And then to prove that
these two are congruent,

00:02:40.970 --> 00:02:43.500
we use the exact same logic.

00:02:43.500 --> 00:02:47.445
So first of all, we view
this as a transversal.

00:02:47.445 --> 00:02:51.800
We view AC as a
transversal of AB and CD.

00:02:51.800 --> 00:02:56.870
And let me go here and let
me create another point here.

00:02:56.870 --> 00:02:59.140
Let me call this point
F right over here.

00:02:59.140 --> 00:03:05.400
So we know that
angle ACD is going

00:03:05.400 --> 00:03:12.360
to be congruent to
angle FAC because they

00:03:12.360 --> 00:03:16.960
are alternate interior angles.

00:03:16.960 --> 00:03:19.115
And then we change our
thinking a little bit.

00:03:19.115 --> 00:03:21.990
And we view AC and BD
as the parallel lines

00:03:21.990 --> 00:03:24.020
and AB as a transversal.

00:03:24.020 --> 00:03:31.260
And then angle FAC is going
to be congruent to angle ABD,

00:03:31.260 --> 00:03:33.430
because they're
corresponding angles.

00:03:33.430 --> 00:03:40.390
Angle F to angle ABD, and
they are corresponding angles.

00:03:40.390 --> 00:03:43.630
So in the first time, we viewed
this as the transversal, AC

00:03:43.630 --> 00:03:47.810
as a transversal of AB and
CD, which are parallel lines.

00:03:47.810 --> 00:03:52.770
Now AB is the transversal and BD
and AC are the parallel lines.

00:03:52.770 --> 00:03:54.666
And obviously, if this
is congruent to that,

00:03:54.666 --> 00:03:56.540
and that is congruent
to that, then these two

00:03:56.540 --> 00:03:58.280
have to be congruent
to each other.

00:03:58.280 --> 00:04:02.030
So we see that if we have
opposite angles are congruent--

00:04:02.030 --> 00:04:04.320
or if we have a
parallelogram, then

00:04:04.320 --> 00:04:08.050
the opposite angles are
going to be congruent.


Proof: Diagonals of a parallelogram bisect each other

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.560
So we have a parallelogram
right over here.

00:00:02.560 --> 00:00:04.880
And what I want to prove
is that its diagonals

00:00:04.880 --> 00:00:06.717
bisect each other.

00:00:06.717 --> 00:00:08.800
So the first thing that
we can think about-- these

00:00:08.800 --> 00:00:10.050
aren't just diagonals.

00:00:10.050 --> 00:00:12.430
These are lines that are
intersecting, parallel lines.

00:00:12.430 --> 00:00:14.660
So you can also view
them as transversals.

00:00:14.660 --> 00:00:17.280
And if we focus on
DB right over here,

00:00:17.280 --> 00:00:21.090
we see that it
intersects DC and AB.

00:00:21.090 --> 00:00:24.825
And since we know that
they're parallel--

00:00:24.825 --> 00:00:26.200
this is a
parallelogram-- we know

00:00:26.200 --> 00:00:29.060
the alternate interior
angles must be congruent.

00:00:29.060 --> 00:00:31.140
So that angle must be
equal to that angle there.

00:00:31.140 --> 00:00:32.650
And let me make a label here.

00:00:32.650 --> 00:00:34.450
Let me call that
middle point E. So we

00:00:34.450 --> 00:00:45.680
know that angle ABE must
be congruent to angle CDE

00:00:45.680 --> 00:00:51.320
by alternate interior angles
of a transversal intersecting

00:00:51.320 --> 00:00:52.200
parallel lines.

00:00:56.930 --> 00:00:59.430
Now, if we look at
diagonal AC-- or we

00:00:59.430 --> 00:01:02.960
should call it transversal AC--
we can make the same argument.

00:01:02.960 --> 00:01:04.769
It intersects here and here.

00:01:04.769 --> 00:01:06.510
These two lines are parallel.

00:01:06.510 --> 00:01:09.320
So alternate interior
angles must be congruent.

00:01:09.320 --> 00:01:13.290
So angle DEC must be-- so let
me write this down-- angle

00:01:13.290 --> 00:01:27.284
DEC must be congruent to angle
BAE, for the exact same reason.

00:01:27.284 --> 00:01:28.700
Now we have something
interesting,

00:01:28.700 --> 00:01:30.408
if we look at this
top triangle over here

00:01:30.408 --> 00:01:31.880
and this bottom triangle.

00:01:31.880 --> 00:01:36.260
We have one set of corresponding
angles that are congruent.

00:01:36.260 --> 00:01:39.217
We have a side in between
that's going to be congruent.

00:01:39.217 --> 00:01:41.050
Actually, let me write
that down explicitly.

00:01:41.050 --> 00:01:44.740
We know-- and we proved
this to ourselves

00:01:44.740 --> 00:01:48.560
in the previous video-- that
parallelograms-- not only are

00:01:48.560 --> 00:01:51.530
opposite sides parallel,
they are also congruent.

00:01:51.530 --> 00:01:53.040
So we know from
the previous video

00:01:53.040 --> 00:01:55.340
that that side is
equal to that side.

00:01:55.340 --> 00:01:57.150
So let me go back to
what I was saying.

00:01:57.150 --> 00:01:58.875
We have two sets of
corresponding angles

00:01:58.875 --> 00:02:01.834
that are congruent, we
have a side in between

00:02:01.834 --> 00:02:03.750
that's congruent, and
then we have another set

00:02:03.750 --> 00:02:05.692
of corresponding angles
that are congruent.

00:02:05.692 --> 00:02:08.150
So we know that this triangle
is congruent to that triangle

00:02:08.150 --> 00:02:08.983
by angle-side-angle.

00:02:12.070 --> 00:02:14.670
So we know that
triangle-- I'm going

00:02:14.670 --> 00:02:17.500
to go from the blue to the
orange to the last one--

00:02:17.500 --> 00:02:23.150
triangle ABE is congruent to
triangle-- blue, orange, then

00:02:23.150 --> 00:02:29.480
the last one-- CDE, by
angle-side-angle congruency.

00:02:34.010 --> 00:02:35.930
Now, what does that do for us?

00:02:35.930 --> 00:02:38.270
Well, we know if two
triangles are congruent,

00:02:38.270 --> 00:02:39.897
all of their
corresponding features,

00:02:39.897 --> 00:02:41.730
especially all of their
corresponding sides,

00:02:41.730 --> 00:02:42.850
are congruent.

00:02:42.850 --> 00:02:49.970
So we know that side EC
corresponds to side EA.

00:02:49.970 --> 00:03:00.806
Or I could say side AE
corresponds to side CE.

00:03:00.806 --> 00:03:02.930
They're corresponding sides
of congruent triangles,

00:03:02.930 --> 00:03:05.380
so their measures or their
lengths must be the same.

00:03:05.380 --> 00:03:09.207
So AE must be equal to CE.

00:03:09.207 --> 00:03:10.790
Let me put two slashes
since I already

00:03:10.790 --> 00:03:12.630
used one slash over here.

00:03:12.630 --> 00:03:15.300
Now, by the same
exact logic, we know

00:03:15.300 --> 00:03:19.060
that DE-- let me
focus on this-- we

00:03:19.060 --> 00:03:26.100
know that BE must
be equal to DE.

00:03:26.100 --> 00:03:27.910
Once again, they're
corresponding sides

00:03:27.910 --> 00:03:31.280
of two congruent triangles, so
they must have the same length.

00:03:31.280 --> 00:03:38.440
So this is corresponding
sides of congruent triangles.

00:03:38.440 --> 00:03:42.745
So BE is equal to DE.

00:03:42.745 --> 00:03:44.150
And we've done our proof.

00:03:44.150 --> 00:03:48.220
We've shown that, look,
diagonal DB is splitting AC

00:03:48.220 --> 00:03:51.420
into two segments of equal
length and vice versa.

00:03:51.420 --> 00:03:55.550
AC is splitting DB into two
segments of equal length.

00:03:55.550 --> 00:03:58.224
So they are
bisecting each other.

00:03:58.224 --> 00:03:59.640
Now let's go the
other way around.

00:03:59.640 --> 00:04:02.930
Let's prove to
ourselves that if we

00:04:02.930 --> 00:04:04.870
have two diagonals of
a quadrilateral that

00:04:04.870 --> 00:04:06.900
are bisecting each
other, that we

00:04:06.900 --> 00:04:09.180
are dealing with
a parallelogram.

00:04:09.180 --> 00:04:09.890
So let me see.

00:04:09.890 --> 00:04:12.630
So we're going to assume that
the two diagonals are bisecting

00:04:12.630 --> 00:04:13.140
each other.

00:04:13.140 --> 00:04:15.130
So we're assuming that
that is equal to that

00:04:15.130 --> 00:04:17.519
and that that right over
there is equal to that.

00:04:17.519 --> 00:04:22.690
Given that, we want to prove
that this is a parallelogram.

00:04:22.690 --> 00:04:28.190
And to do that, we just
have to remind ourselves

00:04:28.190 --> 00:04:30.566
that this angle is going to
be equal to that angle-- it's

00:04:30.566 --> 00:04:32.773
one of the first things we
learned-- because they are

00:04:32.773 --> 00:04:33.720
vertical angles.

00:04:33.720 --> 00:04:35.630
So let me write this down.

00:04:35.630 --> 00:04:38.540
Let me label this point.

00:04:38.540 --> 00:04:43.730
Angle CED is going
to be equal to--

00:04:43.730 --> 00:04:47.385
or is congruent to-- angle BEA.

00:04:53.620 --> 00:04:55.454
Well, that shows us
that these two triangles

00:04:55.454 --> 00:04:57.869
are congruent because we have
our corresponding sides that

00:04:57.869 --> 00:05:00.560
are congruent, an angle in
between, and then another side.

00:05:00.560 --> 00:05:02.730
So we now know that
triangle-- I'll

00:05:02.730 --> 00:05:07.910
keep this in
yellow-- triangle AEB

00:05:07.910 --> 00:05:22.590
is congruent to triangle DEC
by side-angle-side congruency,

00:05:22.590 --> 00:05:28.320
by SAS congruent triangles.

00:05:28.320 --> 00:05:29.019
Fair enough.

00:05:29.019 --> 00:05:31.060
Now, if we know that two
triangles are congruent,

00:05:31.060 --> 00:05:33.310
we know that all of the
corresponding sides and angles

00:05:33.310 --> 00:05:34.470
are congruent.

00:05:34.470 --> 00:05:42.910
So for example, we
know that angle CDE

00:05:42.910 --> 00:05:47.240
is going to be
congruent to angle BAE.

00:05:55.660 --> 00:06:06.140
And this is just corresponding
angles of congruent triangles.

00:06:06.140 --> 00:06:12.310
And now we have this
transversal of these two lines

00:06:12.310 --> 00:06:16.030
that could be parallel, if the
alternate interior angles are

00:06:16.030 --> 00:06:16.640
congruent.

00:06:16.640 --> 00:06:18.150
And we see that they are.

00:06:18.150 --> 00:06:22.440
These two are kind of candidate
alternate interior angles,

00:06:22.440 --> 00:06:24.070
and they are congruent.

00:06:24.070 --> 00:06:28.072
So AB must be parallel to CD.

00:06:28.072 --> 00:06:29.530
Actually, I'll just
draw one arrow.

00:06:33.080 --> 00:06:39.570
AB is parallel to CD by
alternate interior angles

00:06:39.570 --> 00:06:42.932
congruent of parallel lines.

00:06:42.932 --> 00:06:44.390
I'm just writing
in some shorthand.

00:06:44.390 --> 00:06:46.360
Forgive the cryptic
nature of it.

00:06:46.360 --> 00:06:47.910
I'm saying it out.

00:06:47.910 --> 00:06:49.825
And so we can then
do the exact same--

00:06:49.825 --> 00:06:51.950
we've just shown that these
two sides are parallel.

00:06:51.950 --> 00:06:53.491
We could then do
the exact same logic

00:06:53.491 --> 00:06:55.262
to show that these two
sides are parallel.

00:06:55.262 --> 00:06:56.970
And I won't necessarily
write it all out,

00:06:56.970 --> 00:07:00.220
but it's the exact same
proof to show that these two.

00:07:00.220 --> 00:07:01.920
So first of all, we
know that this angle

00:07:01.920 --> 00:07:05.300
is congruent to that
angle right over there.

00:07:05.300 --> 00:07:06.550
Actually, let me write it out.

00:07:06.550 --> 00:07:16.990
So we know that angle AEC
is congruent to angle DEB.

00:07:22.840 --> 00:07:24.020
They are vertical angles.

00:07:26.890 --> 00:07:28.710
And that was our reason
up here, as well.

00:07:32.020 --> 00:07:34.530
And then we see the
triangle AEC must

00:07:34.530 --> 00:07:38.630
be congruent to triangle
DEB by side-angle-side.

00:07:38.630 --> 00:07:43.330
So then we have
triangle AEC must

00:07:43.330 --> 00:07:50.950
be congruent to triangle
DEB by SAS congruency.

00:07:50.950 --> 00:07:55.520
Then we know that corresponding
angles must be congruent.

00:07:55.520 --> 00:08:03.680
So for example, angle CAE must
be congruent to angle BDE.

00:08:09.350 --> 00:08:11.840
And this is they're
corresponding angles

00:08:11.840 --> 00:08:13.560
of congruent triangles.

00:08:13.560 --> 00:08:21.940
So CAE-- let me do
this in a new color--

00:08:21.940 --> 00:08:23.435
must be congruent to BDE.

00:08:28.290 --> 00:08:30.080
And now we have a transversal.

00:08:30.080 --> 00:08:32.280
The alternate interior
angles are congruent.

00:08:32.280 --> 00:08:34.659
So the two lines that the
transversal is intersecting

00:08:34.659 --> 00:08:36.360
must be parallel.

00:08:36.360 --> 00:08:39.130
So this must be
parallel to that.

00:08:39.130 --> 00:08:43.750
So then we have AC
must be parallel to be

00:08:43.750 --> 00:08:47.445
BD by alternate interior angles.

00:08:50.665 --> 00:08:51.290
And we're done.

00:08:51.290 --> 00:08:53.530
We've just proven that
if the diagonals bisect

00:08:53.530 --> 00:08:57.150
each other, if we start that as
a given, then we end at a point

00:08:57.150 --> 00:08:59.650
where we say, hey, the opposite
sides of this quadrilateral

00:08:59.650 --> 00:09:05.030
must be parallel, or that
ABCD is a parallelogram.


Proof: Opposite sides of parallelogram congruent

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.040
What we're going to
prove in this video

00:00:03.040 --> 00:00:05.300
is a couple of fairly
straightforward

00:00:05.300 --> 00:00:06.837
parallelogram-related proofs.

00:00:06.837 --> 00:00:08.670
And this first one,
we're going to say, hey,

00:00:08.670 --> 00:00:10.720
if we have this
parallelogram ABCD,

00:00:10.720 --> 00:00:13.920
let's prove that the opposite
sides have the same length.

00:00:13.920 --> 00:00:19.820
So prove that AB is equal to
DC and that AD is equal to BC.

00:00:19.820 --> 00:00:21.630
So let me draw a diagonal here.

00:00:24.740 --> 00:00:26.850
And this diagonal, depending
on how you view it,

00:00:26.850 --> 00:00:29.900
is intersecting two
sets of parallel lines.

00:00:29.900 --> 00:00:32.150
So you could also consider
it to be a transversal.

00:00:32.150 --> 00:00:34.441
Actually, let me draw it a
little bit neater than that.

00:00:34.441 --> 00:00:35.790
I can do a better job.

00:00:35.790 --> 00:00:36.290
Nope.

00:00:36.290 --> 00:00:38.550
That's not any better.

00:00:38.550 --> 00:00:41.530
That is about as
good as I can do.

00:00:41.530 --> 00:00:44.000
So if we view DB,
this diagonal DB--

00:00:44.000 --> 00:00:47.780
we can view it as a transversal
for the parallel lines AB

00:00:47.780 --> 00:00:49.110
and DC.

00:00:49.110 --> 00:00:51.800
And if you view it that
way, you can pick out

00:00:51.800 --> 00:00:56.484
that angle ABD is going to
be congruent-- so angle ABD.

00:00:56.484 --> 00:00:57.900
That's that angle
right there-- is

00:00:57.900 --> 00:01:00.960
going to be congruent
to angle BDC,

00:01:00.960 --> 00:01:03.430
because they are
alternate interior angles.

00:01:03.430 --> 00:01:05.300
You have a transversal--
parallel lines.

00:01:05.300 --> 00:01:09.810
So we know that
angle ABD is going

00:01:09.810 --> 00:01:12.007
to be congruent to angle BDC.

00:01:16.020 --> 00:01:18.360
Now, you could also
view this diagonal,

00:01:18.360 --> 00:01:21.490
DB-- you could view it as
a transversal of these two

00:01:21.490 --> 00:01:26.030
parallel lines, of the other
pair of parallel lines, AD

00:01:26.030 --> 00:01:27.600
and BC.

00:01:27.600 --> 00:01:29.990
And if you look at it that
way, then you immediately

00:01:29.990 --> 00:01:38.130
see that angle DBC
right over here

00:01:38.130 --> 00:01:40.490
is going to be
congruent to angle

00:01:40.490 --> 00:01:47.690
ADB for the exact same reason.

00:01:47.690 --> 00:01:49.630
They are alternate
interior angles

00:01:49.630 --> 00:01:53.170
of a transversal intersecting
these two parallel lines.

00:01:53.170 --> 00:01:54.360
So I could write this.

00:01:54.360 --> 00:02:00.810
This is alternate
interior angles

00:02:00.810 --> 00:02:04.730
are congruent when you have
a transversal intersecting

00:02:04.730 --> 00:02:06.880
two parallel lines.

00:02:06.880 --> 00:02:09.520
And we also see that
both of these triangles,

00:02:09.520 --> 00:02:15.910
triangle ADB and triangle CDB,
both share this side over here.

00:02:15.910 --> 00:02:18.440
It's obviously equal to itself.

00:02:18.440 --> 00:02:20.425
Now, why is this useful?

00:02:20.425 --> 00:02:22.050
Well, you might
realize that we've just

00:02:22.050 --> 00:02:23.820
shown that both of
these triangles,

00:02:23.820 --> 00:02:25.340
they have this pink angle.

00:02:25.340 --> 00:02:27.532
Then they have this
side in common.

00:02:27.532 --> 00:02:28.990
And then they have
the green angle.

00:02:28.990 --> 00:02:32.580
Pink angle, side in common,
and then the green angle.

00:02:32.580 --> 00:02:35.470
So we've just shown
by angle-side-angle

00:02:35.470 --> 00:02:38.040
that these two
triangles are congruent.

00:02:38.040 --> 00:02:39.440
So let me write this down.

00:02:39.440 --> 00:02:42.320
We have shown that
triangle-- I'll

00:02:42.320 --> 00:02:47.650
go from non-labeled
to pink to green-- ADB

00:02:47.650 --> 00:02:53.105
is congruent to triangle--
non-labeled to pink to green--

00:02:53.105 --> 00:02:53.605
CBD.

00:02:58.810 --> 00:03:02.440
And this comes out of
angle-side-angle congruency.

00:03:09.390 --> 00:03:11.190
Well, what does that do for us?

00:03:11.190 --> 00:03:13.360
Well, if two triangles
are congruent, then

00:03:13.360 --> 00:03:16.350
all of the corresponding
features of the two triangles

00:03:16.350 --> 00:03:18.010
are going to be congruent.

00:03:18.010 --> 00:03:26.360
In particular, side DC
on this bottom triangle

00:03:26.360 --> 00:03:29.110
corresponds to side BA
on that top triangle.

00:03:29.110 --> 00:03:32.620
So they need to be congruent.

00:03:32.620 --> 00:03:38.230
So we get DC is going
to be equal to BA.

00:03:38.230 --> 00:03:43.370
And that's because they
are corresponding sides

00:03:43.370 --> 00:03:47.200
of congruent triangles.

00:03:47.200 --> 00:03:49.560
So this is going to
be equal to that.

00:03:49.560 --> 00:03:54.190
And by that exact same
logic, AD corresponds to CB.

00:03:58.260 --> 00:04:00.770
AD is equal to CB.

00:04:00.770 --> 00:04:03.680
And for the exact same
reason-- corresponding sides

00:04:03.680 --> 00:04:05.240
of congruent triangles.

00:04:05.240 --> 00:04:06.740
And then we're done.

00:04:06.740 --> 00:04:09.896
We've proven that opposite
sides are congruent.

00:04:09.896 --> 00:04:11.020
Now let's go the other way.

00:04:13.640 --> 00:04:16.269
Let's say that we have some
type of a quadrilateral,

00:04:16.269 --> 00:04:18.829
and we know that the
opposite sides are congruent.

00:04:18.829 --> 00:04:22.460
Can we prove to ourselves
that this is a parallelogram?

00:04:22.460 --> 00:04:24.900
Well, it's kind of the
same proof in reverse.

00:04:24.900 --> 00:04:27.080
So let's draw a
diagonal here, since we

00:04:27.080 --> 00:04:29.250
know a lot about triangles.

00:04:29.250 --> 00:04:31.760
So let me draw.

00:04:31.760 --> 00:04:34.200
There we go.

00:04:34.200 --> 00:04:35.336
That's the hardest part.

00:04:35.336 --> 00:04:36.410
Draw it.

00:04:36.410 --> 00:04:37.930
That's pretty good.

00:04:37.930 --> 00:04:38.430
All right.

00:04:38.430 --> 00:04:42.710
So we obviously know that CB
is going to be equal to itself.

00:04:42.710 --> 00:04:44.600
So I'll draw it like that.

00:04:44.600 --> 00:04:46.695
Obviously, because
it's the same line.

00:04:46.695 --> 00:04:48.320
And then we have
something interesting.

00:04:48.320 --> 00:04:51.500
We've split this quadrilateral
into two triangles, triangle

00:04:51.500 --> 00:04:56.500
ACB and triangle DBC.

00:04:56.500 --> 00:05:00.430
And notice, all three sides
of these two triangles

00:05:00.430 --> 00:05:01.650
are equal to each other.

00:05:01.650 --> 00:05:05.110
So we know by side-side-side
that they are congruent.

00:05:05.110 --> 00:05:11.125
So we know that triangle
A-- and we're starting at A,

00:05:11.125 --> 00:05:13.380
and then I'm going
to the one-hash side.

00:05:13.380 --> 00:05:20.263
So ACB is congruent
to triangle DBC.

00:05:24.030 --> 00:05:30.670
And this is by
side-side-side congruency.

00:05:30.670 --> 00:05:32.300
Well, what does that do for us?

00:05:32.300 --> 00:05:34.550
Well, it tells us that all
of the corresponding angles

00:05:34.550 --> 00:05:36.430
are going to be congruent.

00:05:36.430 --> 00:05:42.200
So for example, angle ABC
is going to be-- so let

00:05:42.200 --> 00:05:43.830
me mark that.

00:05:49.160 --> 00:05:51.780
You can say ABC is going
to be congruent to DCB.

00:05:57.480 --> 00:06:04.300
And you could say, by
corresponding angles congruent

00:06:04.300 --> 00:06:06.560
of congruent triangles.

00:06:06.560 --> 00:06:09.200
I'm just using some shorthand
here to save some time.

00:06:09.200 --> 00:06:12.080
So ABC is going to
be congruent to DCB,

00:06:12.080 --> 00:06:15.540
so these two angles are
going to be congruent.

00:06:15.540 --> 00:06:18.360
Well, this is interesting,
because here you have a line.

00:06:18.360 --> 00:06:21.460
And it's intersecting AB and CD.

00:06:21.460 --> 00:06:23.970
And we clearly see
that these things that

00:06:23.970 --> 00:06:27.720
could be alternate interior
angles are congruent.

00:06:27.720 --> 00:06:30.360
And because we have these
congruent alternate interior

00:06:30.360 --> 00:06:34.200
angles, we know that AB
must be parallel to CD.

00:06:34.200 --> 00:06:36.710
So this must be
parallel to that.

00:06:36.710 --> 00:06:41.630
So we know that AB
is parallel to CD

00:06:41.630 --> 00:06:50.350
by alternate interior angles
of a transversal intersecting

00:06:50.350 --> 00:06:51.730
parallel lines.

00:06:51.730 --> 00:06:53.840
Now, we can use that
exact same logic.

00:06:53.840 --> 00:06:56.720
We also know that angle--
let me get this right.

00:06:56.720 --> 00:07:03.785
Angle ACB is congruent
to angle DBC.

00:07:09.710 --> 00:07:16.170
And we know that by
corresponding angles congruent

00:07:16.170 --> 00:07:18.790
of congruent triangles.

00:07:18.790 --> 00:07:22.580
So we're just saying this
angle is equal to that angle.

00:07:22.580 --> 00:07:25.052
Well, once again, these could
be alternate interior angles.

00:07:25.052 --> 00:07:26.260
They look like they could be.

00:07:26.260 --> 00:07:27.280
This is a transversal.

00:07:27.280 --> 00:07:28.980
And here's two lines
here, which we're not sure

00:07:28.980 --> 00:07:30.100
whether they're parallel.

00:07:30.100 --> 00:07:33.030
But because the alternate
interior angles are congruent,

00:07:33.030 --> 00:07:34.990
we know that they are parallel.

00:07:34.990 --> 00:07:36.860
So this is parallel to that.

00:07:36.860 --> 00:07:41.500
So we know that AC
is parallel to BD

00:07:41.500 --> 00:07:43.960
by alternate interior angles.

00:07:48.660 --> 00:07:49.610
And we're done.

00:07:49.610 --> 00:07:51.390
So what we've done
is-- it's interesting.

00:07:51.390 --> 00:07:55.680
We've shown if you
have a parallelogram,

00:07:55.680 --> 00:07:57.640
opposite sides have
the same length.

00:07:57.640 --> 00:07:59.670
And if opposite sides
have the same length,

00:07:59.670 --> 00:08:01.030
then you have a parallelogram.

00:08:01.030 --> 00:08:03.340
And so we've actually proven
it in both directions.

00:08:03.340 --> 00:08:06.080
And so we can actually make what
you call an "if and only if"

00:08:06.080 --> 00:08:06.580
statement.

00:08:12.150 --> 00:08:16.020
You could say opposite sides of
a quadrilateral are parallel if

00:08:16.020 --> 00:08:18.520
and only if their
lengths are equal.

00:08:18.520 --> 00:08:20.075
And you say if and only if.

00:08:20.075 --> 00:08:21.624
So if they are
parallel, then you

00:08:21.624 --> 00:08:23.040
could say their
lengths are equal.

00:08:23.040 --> 00:08:26.410
And only if their lengths
are equal are they parallel.

00:08:26.410 --> 00:08:28.875
We've proven it in
both directions.


Quadrilateral overview

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:01.830
What I want to do
in this video is

00:00:01.830 --> 00:00:05.030
give an overview
of quadrilaterals.

00:00:05.030 --> 00:00:07.531
And you can imagine,
from this prefix,

00:00:07.531 --> 00:00:09.780
or, I guess you could say,
the beginning of this word,

00:00:09.780 --> 00:00:12.560
quad-- this involves
four of something.

00:00:12.560 --> 00:00:15.244
And quadrilaterals, as you
can imagine, are shapes.

00:00:15.244 --> 00:00:17.660
And we're going to be talking
about two-dimensional shapes

00:00:17.660 --> 00:00:21.260
that have four sides and four
vertices and four angles.

00:00:21.260 --> 00:00:25.701
So, for example--
one, two, three, four.

00:00:25.701 --> 00:00:27.700
That is a quadrilateral,
although that last side

00:00:27.700 --> 00:00:29.040
didn't look too straight.

00:00:29.040 --> 00:00:32.509
One, two, three, four.

00:00:32.509 --> 00:00:33.950
That is a quadrilateral.

00:00:33.950 --> 00:00:35.492
One, two, three, four.

00:00:35.492 --> 00:00:36.700
These are all quadrilaterals.

00:00:36.700 --> 00:00:39.400
They all have four
sides, four vertices,

00:00:39.400 --> 00:00:40.820
and, clearly, four angles.

00:00:40.820 --> 00:00:45.836
One angle, two angles, three
angles, and four angles.

00:00:45.836 --> 00:00:47.960
Actually, let me draw this
one a little bit bigger,

00:00:47.960 --> 00:00:49.370
because it's interesting.

00:00:49.370 --> 00:00:51.440
So in this one
right over here, you

00:00:51.440 --> 00:00:53.900
have one angle, two angles,
three angles, and then

00:00:53.900 --> 00:00:56.440
you have this really big
angle right over there.

00:00:56.440 --> 00:01:00.800
If you look at the interior
angles of this quadrilateral.

00:01:00.800 --> 00:01:02.800
Now, quadrilaterals,
as you can imagine,

00:01:02.800 --> 00:01:05.680
can be subdivided
into other groups

00:01:05.680 --> 00:01:07.970
based on the properties
of the quadrilaterals.

00:01:07.970 --> 00:01:11.120
And the main subdivision
of quadrilaterals

00:01:11.120 --> 00:01:14.540
is between concave and
convex quadrilaterals.

00:01:14.540 --> 00:01:19.870
So you have concave,
and you have convex.

00:01:19.870 --> 00:01:22.490
And the way I remember
concave quadrilaterals,

00:01:22.490 --> 00:01:25.860
or really concave polygons
of any number of shapes,

00:01:25.860 --> 00:01:28.490
is that it looks like
something has caved in.

00:01:28.490 --> 00:01:30.900
So, for example, this is
a concave quadrilateral.

00:01:30.900 --> 00:01:33.820
It looks like this
side has been caved in.

00:01:33.820 --> 00:01:37.270
And one way to define
concave quadrilaterals--

00:01:37.270 --> 00:01:40.410
so let me draw it a little bit
bigger, so this right over here

00:01:40.410 --> 00:01:43.550
is a concave
quadrilateral-- is that it

00:01:43.550 --> 00:01:49.400
has an interior angle that
is larger than 180 degrees.

00:01:49.400 --> 00:01:52.230
So for example, this interior
angle right over here

00:01:52.230 --> 00:01:57.070
is larger than 180 degrees.

00:01:57.070 --> 00:01:58.480
And it's an interesting proof.

00:01:58.480 --> 00:01:59.396
Maybe I'll do a video.

00:01:59.396 --> 00:02:01.450
It's actually a pretty
simple proof to show that,

00:02:01.450 --> 00:02:04.170
if you have a concave
quadrilateral, if at least one

00:02:04.170 --> 00:02:07.390
of the interior angles has
a measure larger than 180

00:02:07.390 --> 00:02:10.510
degrees, that none of the sides
can be parallel to each other.

00:02:10.510 --> 00:02:12.900
The other type of
quadrilateral, you can imagine,

00:02:12.900 --> 00:02:14.690
is when all of the
interior angles

00:02:14.690 --> 00:02:17.013
are less than 180 degrees.

00:02:17.013 --> 00:02:19.263
And you might say, wait--
what happens at 180 degrees?

00:02:19.263 --> 00:02:20.939
Well, if this angle
was 180 degrees,

00:02:20.939 --> 00:02:22.730
then these wouldn't be
two different sides,

00:02:22.730 --> 00:02:23.740
it would just be one side.

00:02:23.740 --> 00:02:25.290
And that would look
like a triangle.

00:02:25.290 --> 00:02:26.800
But if all of the
interior angles

00:02:26.800 --> 00:02:29.670
are less than 180
degrees, then you're

00:02:29.670 --> 00:02:32.820
dealing with a
convex quadrilateral.

00:02:32.820 --> 00:02:35.890
So this convex
quadrilateral would

00:02:35.890 --> 00:02:37.780
involve that one and
that one over there.

00:02:37.780 --> 00:02:44.840
So this right over here is
what a convex quadrilateral

00:02:44.840 --> 00:02:49.600
could look like-- four points,
four sides, four angles.

00:02:49.600 --> 00:02:52.090
Now, within convex
quadrilaterals,

00:02:52.090 --> 00:02:55.330
there are some other
interesting categorizations.

00:02:55.330 --> 00:02:59.230
So now we're just going to
focus on convex quadrilaterals,

00:02:59.230 --> 00:03:01.510
so that's going to be all
of this space over here.

00:03:01.510 --> 00:03:07.200
So one type of convex
quadrilateral is a trapezoid.

00:03:07.200 --> 00:03:09.280
And a trapezoid is a
convex quadrilateral,

00:03:09.280 --> 00:03:10.740
and sometimes the
definition here

00:03:10.740 --> 00:03:12.400
is a little bit--
different people will

00:03:12.400 --> 00:03:13.770
use different definitions.

00:03:13.770 --> 00:03:16.480
So some people will
say a trapezoid

00:03:16.480 --> 00:03:20.580
is a quadrilateral that
has exactly two sides that

00:03:20.580 --> 00:03:21.850
are parallel to each other.

00:03:21.850 --> 00:03:26.540
So, for example, they would
say that this right over here

00:03:26.540 --> 00:03:31.350
is a trapezoid, where this
side is parallel to that side.

00:03:31.350 --> 00:03:38.600
If I give it some letters here,
if I call this trapezoid ABCD,

00:03:38.600 --> 00:03:43.750
we could say that segment AB
is parallel to segment DC,

00:03:43.750 --> 00:03:48.349
and because of that we know
that this is a trapezoid.

00:03:48.349 --> 00:03:50.390
Now I said that the
definition is a little fuzzy,

00:03:50.390 --> 00:03:53.820
because some people say
you can have exactly one

00:03:53.820 --> 00:03:56.070
pair of parallel
sides, but some people

00:03:56.070 --> 00:03:58.590
say at least one pair
of parallel sides.

00:03:58.590 --> 00:04:00.222
So if you use the
original definition--

00:04:00.222 --> 00:04:02.680
and that's the kind of thing
that most people are referring

00:04:02.680 --> 00:04:04.820
to when they say a
trapezoid, exactly

00:04:04.820 --> 00:04:07.560
one pair of parallel sides-- It
might be something like this.

00:04:07.560 --> 00:04:11.550
But if you use the broader
definition of at least one pair

00:04:11.550 --> 00:04:14.380
of parallel sides,
then maybe this

00:04:14.380 --> 00:04:16.320
could also be
considered a trapezoid

00:04:16.320 --> 00:04:18.600
so you have one pair of
parallel sides like that

00:04:18.600 --> 00:04:21.630
and then you have another pair
of parallel sides like that.

00:04:21.630 --> 00:04:24.840
So this is a question mark
where it comes to a trapezoid.

00:04:24.840 --> 00:04:26.740
A trapezoid is definitely
this thing here,

00:04:26.740 --> 00:04:29.160
where you have exactly one
pair of parallel sides.

00:04:29.160 --> 00:04:31.130
Depending on
people's definition,

00:04:31.130 --> 00:04:32.916
this may or may
not be a trapezoid.

00:04:32.916 --> 00:04:35.040
If you say it's exactly
one pair of parallel sides,

00:04:35.040 --> 00:04:37.810
this is not a trapezoid,
because it has two pairs.

00:04:37.810 --> 00:04:39.770
If you say at least one
pair of parallel sides,

00:04:39.770 --> 00:04:40.930
then this is a trapezoid.

00:04:40.930 --> 00:04:43.270
So I'll put that in a
little question mark there.

00:04:43.270 --> 00:04:45.260
But there is a name
for this, regardless

00:04:45.260 --> 00:04:47.560
of your definition of
what a trapezoid is.

00:04:47.560 --> 00:04:50.990
If you have a quadrilateral with
two pairs of parallel sides,

00:04:50.990 --> 00:04:53.300
you are then dealing
with a parallelogram.

00:04:53.300 --> 00:04:55.430
So the one thing that you
definitely can call this

00:04:55.430 --> 00:04:56.523
is a parallelogram.

00:05:06.001 --> 00:05:07.750
And I'll just draw it
a little bit bigger.

00:05:07.750 --> 00:05:11.390
So it's a quadrilateral, and
if I have a quadrilateral,

00:05:11.390 --> 00:05:13.850
and if I have two pairs
of parallel sides.

00:05:13.850 --> 00:05:16.280
So the opposite
sides are parallel.

00:05:16.280 --> 00:05:18.620
So that side is
parallel to that side,

00:05:18.620 --> 00:05:21.930
and then this side is
parallel to that side

00:05:21.930 --> 00:05:24.070
there-- you're dealing
with a parallelogram.

00:05:24.070 --> 00:05:26.890
And then parallelograms can
be subdivided even further.

00:05:30.240 --> 00:05:32.870
If the four angles
in a parallelogram

00:05:32.870 --> 00:05:36.900
are all right angles, you're
dealing with a rectangle.

00:05:36.900 --> 00:05:38.490
So let me draw one like that.

00:05:42.660 --> 00:05:45.276
This is all in the
parallelogram universe,

00:05:45.276 --> 00:05:46.650
what I'm drawing
right over here.

00:05:46.650 --> 00:05:48.686
This is all the
parallelogram universe.

00:05:48.686 --> 00:05:50.310
So it's a parallelogram,
which tells me

00:05:50.310 --> 00:05:52.520
that opposite
sides are parallel.

00:05:52.520 --> 00:05:56.640
And then if we know that all
four angles are 90 degrees.

00:05:56.640 --> 00:05:58.180
And we've proven
in previous videos

00:05:58.180 --> 00:06:01.180
how to figure out the sum of the
interior angles of any polygon.

00:06:01.180 --> 00:06:02.679
And using that same
method you could

00:06:02.679 --> 00:06:07.390
say that the sum of the interior
angles of any quadrilateral

00:06:07.390 --> 00:06:10.130
is actually 360 degrees.

00:06:10.130 --> 00:06:12.230
And you see that in this
special case as well.

00:06:12.230 --> 00:06:14.240
But maybe we'll prove
it in a separate video.

00:06:14.240 --> 00:06:17.810
But this right over here
we would call a rectangle.

00:06:17.810 --> 00:06:19.920
Parallelogram--
opposite sides parallel

00:06:19.920 --> 00:06:22.780
and we have four right angles.

00:06:22.780 --> 00:06:25.520
Now, if we have a parallelogram
where we don't necessarily

00:06:25.520 --> 00:06:29.070
have four right
angles, but where

00:06:29.070 --> 00:06:31.800
we do have the length of
the sides being equal,

00:06:31.800 --> 00:06:34.160
then we're dealing
with a rhombus.

00:06:34.160 --> 00:06:36.100
So let me draw it like that.

00:06:36.100 --> 00:06:37.990
So it's a parallelogram.

00:06:37.990 --> 00:06:42.020
This is a parallelogram, so that
side is parallel to that side,

00:06:42.020 --> 00:06:43.620
this side is parallel
to that side.

00:06:43.620 --> 00:06:46.630
And we also know that all
four sides have equal length.

00:06:46.630 --> 00:06:49.249
So this side's length is equal
to that side's length, which

00:06:49.249 --> 00:06:50.790
is equal to that
side's length, which

00:06:50.790 --> 00:06:52.650
is equal to that side's length.

00:06:52.650 --> 00:06:55.390
Then we are dealing
with a rhombus.

00:06:55.390 --> 00:06:58.190
So one way to view it-- all
rhombi are parallelograms.

00:06:58.190 --> 00:07:00.260
All rectangles are
parallelograms.

00:07:00.260 --> 00:07:03.220
All parallelograms you cannot
assume to be rectangles.

00:07:03.220 --> 00:07:06.720
All parallelograms you
cannot assume to be rhombi.

00:07:06.720 --> 00:07:10.350
Now, something can be both
a rectangle and a rhombus.

00:07:10.350 --> 00:07:13.340
So let's say that this is
the universe of rectangles.

00:07:13.340 --> 00:07:15.260
So the universe of
rectangles-- I'll

00:07:15.260 --> 00:07:17.160
draw a little bit of
a Venn diagram here--

00:07:17.160 --> 00:07:21.070
is that set of shapes and
the universe of rhombi

00:07:21.070 --> 00:07:24.640
is this set of shapes
right over here.

00:07:24.640 --> 00:07:27.050
So what would it look like?

00:07:27.050 --> 00:07:28.880
Well, you would have
four right angles

00:07:28.880 --> 00:07:31.330
and they would all
have the same length.

00:07:31.330 --> 00:07:32.532
So it would look like this.

00:07:32.532 --> 00:07:34.115
So it'd definitely
be a parallelogram.

00:07:36.750 --> 00:07:40.710
Four right angles
and all the sides

00:07:40.710 --> 00:07:41.860
would have the same length.

00:07:41.860 --> 00:07:44.443
And this is probably the first
of the shapes that you learned,

00:07:44.443 --> 00:07:45.670
or one of the first shapes.

00:07:45.670 --> 00:07:48.450
This is clearly a square.

00:07:48.450 --> 00:07:53.904
So all squares could also
be considered a rhombus,

00:07:53.904 --> 00:07:55.820
and they could also be
considered a rectangle,

00:07:55.820 --> 00:07:58.120
and they could also be
considered a parallelogram.

00:07:58.120 --> 00:08:01.240
But clearly, not all
rectangles are squares,

00:08:01.240 --> 00:08:03.530
and not all rhombi are squares.

00:08:03.530 --> 00:08:05.900
And definitely not all
parallelograms are squares.

00:08:05.900 --> 00:08:07.650
This one, clearly,
right over here,

00:08:07.650 --> 00:08:12.000
is neither a rectangle nor
a rhombi, nor a square.

00:08:12.000 --> 00:08:12.940
So that's an overview.

00:08:12.940 --> 00:08:15.882
Just gives you a little bit
of taxonomy of quadrilaterals.

00:08:15.882 --> 00:08:17.340
And then in the
next few videos, we

00:08:17.340 --> 00:08:19.300
can start to explore
them and find

00:08:19.300 --> 00:08:21.270
their interesting
properties or just do

00:08:21.270 --> 00:08:24.250
interesting problems
involving them.


Common orthocenter and centroid

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.350
We're asked to prove
that if the orthocenter

00:00:03.350 --> 00:00:07.420
and centroid of a given
triangle are the same point,

00:00:07.420 --> 00:00:09.052
then the triangle
is equilateral.

00:00:09.052 --> 00:00:10.760
So I have a triangle
over here, and we're

00:00:10.760 --> 00:00:13.910
going to assume that it's
orthocenter and centroid are

00:00:13.910 --> 00:00:14.860
the same point.

00:00:14.860 --> 00:00:17.030
Just as a review,
the orthocenter

00:00:17.030 --> 00:00:19.860
is the point where the three
altitudes of a triangle

00:00:19.860 --> 00:00:23.070
intersect, and the centroid is
a point where the three medians.

00:00:23.070 --> 00:00:25.546
So we can do is we can
assume that these three

00:00:25.546 --> 00:00:26.920
lines right over
here, that these

00:00:26.920 --> 00:00:30.590
are both altitudes and
medians, and that this point

00:00:30.590 --> 00:00:34.400
right over here is both the
orthocenter and the centroid.

00:00:34.400 --> 00:00:36.590
So if we assume that
these lines are altitudes,

00:00:36.590 --> 00:00:39.930
and that tells us that they are
perpendicular to the opposite

00:00:39.930 --> 00:00:41.950
sides-- so that is
a 90 degree angle,

00:00:41.950 --> 00:00:43.610
those are both 90 degree angles.

00:00:43.610 --> 00:00:45.650
These are both 90 degree angles.

00:00:45.650 --> 00:00:47.485
These are both 90 degree angles.

00:00:47.485 --> 00:00:48.860
And the fact that
it's a centroid

00:00:48.860 --> 00:00:51.880
means that each of these lines
bisect at the opposite sides.

00:00:51.880 --> 00:00:55.460
So it tells us that this
length is equal to this length,

00:00:55.460 --> 00:00:58.210
that this length is
a different color,

00:00:58.210 --> 00:01:00.490
this length is equal
to that length,

00:01:00.490 --> 00:01:03.740
and it tells us that
this length-- this length

00:01:03.740 --> 00:01:06.260
is equal to that length.

00:01:06.260 --> 00:01:08.230
And you might also
say, well, gee.

00:01:08.230 --> 00:01:09.690
Each of these lines
are also, then,

00:01:09.690 --> 00:01:12.540
the perpendicular bisectors
of each of the sides.

00:01:12.540 --> 00:01:15.150
So not only is this the
orthocenter in the centroid,

00:01:15.150 --> 00:01:19.092
it is also the circumcenter of
this triangle right over here.

00:01:19.092 --> 00:01:21.300
But with that out of the
way, we've kind of marked up

00:01:21.300 --> 00:01:22.830
everything that we
can assume, given

00:01:22.830 --> 00:01:24.581
that this is an
orthocenter and a center--

00:01:24.581 --> 00:01:27.205
although there are other things,
other properties of especially

00:01:27.205 --> 00:01:28.300
centroids that we know.

00:01:28.300 --> 00:01:29.980
But now let's
prove that this has

00:01:29.980 --> 00:01:32.670
to be an equilateral triangle.

00:01:32.670 --> 00:01:34.900
So the first thing that
you might see-- and let me

00:01:34.900 --> 00:01:36.580
label some letters
here, so we can

00:01:36.580 --> 00:01:38.080
refer to things a
little bit better.

00:01:38.080 --> 00:01:46.820
So let's call that A, B,
C, D, E, and F-- and we

00:01:46.820 --> 00:01:49.930
could label the centroid
right over here, G. So

00:01:49.930 --> 00:01:52.250
the first thing, let's
look at triangle AFG

00:01:52.250 --> 00:01:55.670
and triangle-- with
E-- and triangle EFG.

00:01:55.670 --> 00:02:01.730
So let's compare triangle
A, F-- triangle A, F,

00:02:01.730 --> 00:02:06.540
G-- Let's compare
that to triangle EFG.

00:02:06.540 --> 00:02:09.830
Triangle E,F,G.

00:02:09.830 --> 00:02:11.800
So they definitely
have one side-- side

00:02:11.800 --> 00:02:16.540
EF is congruent to side AF--
these are both the same.

00:02:16.540 --> 00:02:19.790
And we have angle EFG is
the same as angle AFG,

00:02:19.790 --> 00:02:21.190
they're both 90 degrees.

00:02:21.190 --> 00:02:23.520
And then they both
clearly share this side

00:02:23.520 --> 00:02:27.540
FG-- they both share this
side right over here--

00:02:27.540 --> 00:02:29.310
they both share that over there.

00:02:29.310 --> 00:02:32.100
So they have a congruent
side, a corresponding angle

00:02:32.100 --> 00:02:33.690
in between another
congruent side,

00:02:33.690 --> 00:02:35.620
and they're all--
So congruent side,

00:02:35.620 --> 00:02:37.180
congruent corresponding
angle, and

00:02:37.180 --> 00:02:39.500
another congruent
corresponding side.

00:02:39.500 --> 00:02:42.810
So by side, angle, side,
these two characters

00:02:42.810 --> 00:02:44.690
are going to be congruent.

00:02:44.690 --> 00:02:48.750
By side, angle, side congruency.

00:02:48.750 --> 00:02:53.390
Now can you make that
exact same argument

00:02:53.390 --> 00:02:55.790
to say that all of
these pairs that

00:02:55.790 --> 00:02:59.340
have the kind of share-- that
both have these 90 degree

00:02:59.340 --> 00:03:02.730
angles next to each other
are going to be congruent.

00:03:02.730 --> 00:03:07.320
By the same exact argument, we
can say triangle EDG triangle

00:03:07.320 --> 00:03:12.900
E,D,G, is going to be
congruent to triangle CDG.

00:03:12.900 --> 00:03:16.810
Triangle C,D,G. Same exact
thing-- side, angle--

00:03:16.810 --> 00:03:18.869
and then we have a
side right over here.

00:03:18.869 --> 00:03:20.660
And then we can use
the exact same argument

00:03:20.660 --> 00:03:21.618
for this one over here.

00:03:21.618 --> 00:03:28.300
Triangle C-- that looks
like an A-- triangle CBG

00:03:28.300 --> 00:03:31.694
is congruent to triangle ABG.

00:03:31.694 --> 00:03:36.510
A, B, G. That by
itself is interesting.

00:03:36.510 --> 00:03:38.660
But we know that if two
triangles are congruent,

00:03:38.660 --> 00:03:41.120
all of their corresponding
sides and angles

00:03:41.120 --> 00:03:43.230
are going to be congruent.

00:03:43.230 --> 00:03:46.790
So for example, if we know the
measure of this angle is blue,

00:03:46.790 --> 00:03:48.740
the corresponding
angle on this triangle

00:03:48.740 --> 00:03:50.440
is also going to have
the same measure.

00:03:50.440 --> 00:03:52.870
I'll just make it with
that same blue angle.

00:03:52.870 --> 00:03:55.580
And we know if this angle
right over here is magenta,

00:03:55.580 --> 00:03:58.094
the corresponding
angle on triangle AFG

00:03:58.094 --> 00:03:59.760
is also going to have
that same measure.

00:03:59.760 --> 00:04:02.220
And I'll just mark it
with that magenta again.

00:04:02.220 --> 00:04:05.030
Now we also know from our
properties of vertical angles,

00:04:05.030 --> 00:04:07.860
that whatever angle
measure AFG is,

00:04:07.860 --> 00:04:10.630
DGC is going to have the
same measure, because they

00:04:10.630 --> 00:04:12.650
are vertical angles.

00:04:12.650 --> 00:04:14.520
But we know whatever
angle measure

00:04:14.520 --> 00:04:18.769
this is, this
triangle, triangle CDG,

00:04:18.769 --> 00:04:21.990
is congruent to triangle
EDG, so corresponding angles

00:04:21.990 --> 00:04:22.940
have to be congruent.

00:04:22.940 --> 00:04:25.230
So this angle is
this magenta measure,

00:04:25.230 --> 00:04:27.985
then this angle also has
to be the magenta measure.

00:04:27.985 --> 00:04:29.860
And then once again,
you see vertical angles.

00:04:29.860 --> 00:04:31.490
If this is magenta,
then this is also

00:04:31.490 --> 00:04:32.840
going to have that same measure.

00:04:32.840 --> 00:04:34.340
And if this has a
measure, then that

00:04:34.340 --> 00:04:36.131
is also going to have
the same measure.

00:04:36.131 --> 00:04:37.630
So by using a little
bit of argument

00:04:37.630 --> 00:04:39.960
of congruent triangles,
corresponding angles

00:04:39.960 --> 00:04:42.130
are going to be congruent,
and vertical angles,

00:04:42.130 --> 00:04:44.520
we can see that all of these
inner angles right over here

00:04:44.520 --> 00:04:46.640
are going to have to
be the same measure.

00:04:46.640 --> 00:04:48.910
And I'm using that with
this little magenta arc

00:04:48.910 --> 00:04:50.250
right over there.

00:04:50.250 --> 00:04:53.690
Now all of these triangles, that
we split this triangle into,

00:04:53.690 --> 00:04:57.280
they all have a 90 degree angle,
they all have a magenta arc.

00:04:57.280 --> 00:05:01.110
So whatever's left over is
going to be 180 minus 90

00:05:01.110 --> 00:05:02.110
minus magenta.

00:05:02.110 --> 00:05:05.304
Or it's really 90 minus this
magenta angle right over here.

00:05:05.304 --> 00:05:06.970
And that's what this
blue angle must be.

00:05:06.970 --> 00:05:09.890
This blue angle is essentially
90 minus the magenta angle.

00:05:09.890 --> 00:05:12.470
The blue angle is 90
minus the magenta angle.

00:05:12.470 --> 00:05:15.420
And so that angle must be the
third angle for all of these.

00:05:15.420 --> 00:05:17.616
So once again, this
blue angle, there's

00:05:17.616 --> 00:05:19.240
going to be 90 minus
the magenta angle,

00:05:19.240 --> 00:05:22.150
or 180 minus the
magenta minus the 90.

00:05:22.150 --> 00:05:24.200
So that's going to
be a blue angle.

00:05:24.200 --> 00:05:24.970
Essentially what
we're saying is,

00:05:24.970 --> 00:05:26.511
if you know two
angles of a triangle,

00:05:26.511 --> 00:05:29.652
it forces what the other
angle is going to be.

00:05:29.652 --> 00:05:31.610
We already know that all
six of these triangles

00:05:31.610 --> 00:05:33.680
have two angles in common--
the 90 degree angle

00:05:33.680 --> 00:05:34.950
and the magenta angle.

00:05:34.950 --> 00:05:37.390
So the third angle must
be the same as well.

00:05:37.390 --> 00:05:39.700
And we're specifying that
with this blue angle,

00:05:39.700 --> 00:05:41.700
this blue angle right over here.

00:05:41.700 --> 00:05:48.050
And now you see if we
were to look at angle EAC,

00:05:48.050 --> 00:05:53.290
so angle E,A,C. We see that
that's just two of these blue

00:05:53.290 --> 00:05:53.790
angles.

00:05:53.790 --> 00:05:59.470
That is congruent
to angle ACE, A,C,E,

00:05:59.470 --> 00:06:02.250
which is just two of
those blue angles,

00:06:02.250 --> 00:06:04.840
which is congruent to angle CEA.

00:06:04.840 --> 00:06:07.250
Angle C,E,A. Which
is once again,

00:06:07.250 --> 00:06:09.200
just two of those blue angles.

00:06:09.200 --> 00:06:12.290
So we have a triangle here
where the three angles

00:06:12.290 --> 00:06:14.990
in that triangle
all are congruent,

00:06:14.990 --> 00:06:16.910
so it is an
equilateral triangle.

00:06:16.910 --> 00:06:17.930
It's a 60 degree.

00:06:17.930 --> 00:06:20.690
We've proven before if all three
of your angles are the same,

00:06:20.690 --> 00:06:24.840
then the lengths of all three
sides are the same as well.