# A quadrilateral ABCD is formed from four distinct points (called the vertices), no three of which are collinear, and from the segments AB, CB, CD, and DA (called the sides), which have no intersections except at those endpoints labeled by the same letter. The notation for this quadrilateral is not unique- e.g., quadrilateral ABCD = quadrilateral CBAD.

A quadrilateral ABCD is formed from four distinct points (called the vertices), no three of which are collinear, and from the segments AB, CB, CD, and DA (called the sides), which have no intersections except at those endpoints labeled by the same letter. The notation for this quadrilateral is not unique- e.g., quadrilateral $ABCD=$ quadrilateral CBAD.
Two vertices that are endpoints of a side are called adjacent; otherwise the two vertices are called opposite. The remaining pair of segments AC and BD formed from the four points are called diagonals of the quadrilateral; they may or may not intersect at some fifth point. If X, Y, Z are the vertices of quadrilateral ABCD such that Y is adjacent to both X and Z, then angle XYZ is called an angle of the quadrilateral; if W is the fourth vertex, then angle XWZ and angle XYZ are called opposite angles.
The quadrilaterals of main interest are the convex ones. By definition, they are the quadrilaterals such that each pair of opposite sides, e.g., AB and CD, has the property that CD is contained in one of the half-planes bounded by the line through A and B, and AB is contained in one of the half-planes bounded by the line through C and D.
a) Using Pasch's theorem, prove that if one pair of opposite sides has this property, then so does the other pair of opposite sides.
b) Prove, using the crossbar theorem, that the following are equivalent:
2. Each vertex of the quadrilateral lies in the interior of the opposite angle.
3. The diagonals of the quadrilateral meet.
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Bridger Hall
Step 1
In a possible attempt to explain a), let us focus solely on a single angle, say angle A. Similarly, draw tangent lines extending from the two adjacent sides, namely AB and AD. Assuming $A\ne 180$ (which we can, because it would cause ABCD to be a triangle), AB and AD are not parallel.
This means that they meet at A and continue, getting further apart as they go. If $A<180$, meaning ABCD is convex, AB and AD continue away from the shape, not intersecting any sides.
Step 2
However, if $A>180$, AB and AD enter the interior or ABCD after intersecting at A. As the lines are infinite and the quadrilateral is not, the lines must at some point leave the shape. As two lines can only meet at a single point, and will not intersect themselves, they must leave the shape through one of the other two sides (Note Pasch's Theorem).
As both AB and AD are equally dependent on the angle of A, it is not possible for only one of the two lines to split one of the other sides.