Chapter 7 Summary

Misc

General form of indirect proof (also called proof by contradiction): To prove p, assume ~p. Derive from that some contradiction. This disproves ~p, hence proving p.

Definitions

Exterior angle of a polygon

An angle adjacent to and supplementary to an interior angle of the polygon.

Equiangular polygon

A polygon in which all interior angles are congruent

Equilateral polygon

A polygon in which all sides are congruent

Regular polygon

An equilateral and equiangular polygon

Polygon names

(You will only need to know "triangle" through "hexagon" and n-gon.)

Num sides (or vertices)	Name
3	triangle
4	quadrilateral
5	pentagon
6	hexagon
7	heptagon
8	octagon
9	nonagon
10	decagon
12	dodecagon
15	pentadecagon
n	n-gon

Theorems

Th. 50.: The sum of the measures of the three angles of a triangle is 180°.
Th. 51.: The measure of an exterior angle of a triangle equals the sum of the measures of the remote interior angles of the triangle.
Th. 52 ("Midline"): A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is half the length of the third side.
Th. 53 ("No choice"): If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (I.e. there is "no choice" about the measure of the third angle.)
Th. 54 (AAS): If there exists a correspondence between the vertices of two triangles such that two angles and a nonincluded side of one are congruent to the corresponding parts of the other, then the triangles are congruent.

Th. 55

The sum of the measures of the (interior) angles of an n-gon is (n − 2)180°. (To visualize, draw diagonals from one point of the n-gon, count the triangles, and note Theorem 50.)

Th. 56

If an exterior angle is taken at each vertex of an n-gon, the sum of the measures of the exterior angles is 360° (regardless of n). (To visualize, note that if you take a straight angle at each vertex, there are n of them, so their sum is n·180°, and then subtract the sum of the interior angles (Th. 55) to get the sum of the supplements of those interior angles. Those supplements are the exterior angles.)

Th. 57

The number of diagonals in an n-gon is n(n − 3)/2. (To visualize, note that there are n − 3 diagonals from each vertex; there are n vertices, and if you count that way you have counted each diagonal twice.)

Th. 58

The measure of each exterior angle of an equiangular n-gon is 360°/n. (To visualize, just divide the result of Th. 56 by n.)

(No name)

The measure of each interior angle of an equiangular n-gon is 180° − 360°/n. (To visualize, note that the interior angle is the supplement of an exterior angle, and see Th. 58.)