Chapter 7 Final Review

Chapter 7: Polygons

7.1: Triangle Application Theorems

Theorem 50     The sum of the measures of the three angles of a triangle is 180^o

Proof:  According to the Parallel Postulate, there exists exactly one line through point A parallel to line BC, so the figure below can be drawn

Because of the straight angle, we know that < 1+ < 2+ < 3 = 180^o.  Since < 1 is congruent to < B (Parallel lines à alternate interior angles congruent), and < 3 is congruent to < C, we may substitute to obtain < B + < 2 + < C = 180^o Hence, m< A + m< B + m< C = 180^o

Definition        An exterior angle of a polygon is an angle that is adjacent to and supplementary to and interior angle of the polygon. 

Theorem 51     The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Theorem 52     A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one- half the length of the third side.

7.2: Two Proof- Oriented Triangle Theorems

Theorem 53     If two angle of one triangle are congruent to two angles of another triangle, then the third angles are congruent. (No- Choice Theorem)

Since the sum of the angles in each triangle is 180^o, the sums may be set equal.  If we then apply the Subtraction Property, we see that the third angles must be congruent.

Disclaimer: The two Triangles need not be congruent for us to apply the No- Choice Theorem.

Theorem 54     If there exists a correspondence between the vertices of two triangles such that two angles and a non-included side of one arc ore congruent to the corresponding parts of the other, then the triangles are congruent.  (AAS)

                        Really, one uses the No-Choice Theorem to make an AAS into an ASA.

7.3: Formulas Involving Polygons

Theorem 55     The sum S_i of the measures of the angles of a polygon with n sides is given by the formula S_i + (n – 2)180.

On Occasion, we may refer to the angles of a polygon as the interior angles of the polygon.

Theorem 56     If one exterior angle is taken at each vertex, the sum S_e of the measures of the exterior angles of a polygon is given by the formula S_e = 360

Theorem 57     The number d of diagonals that can be drawn in a polygon of n sides is given by the formula:

7.4: Regular Polygons

Definition        A regular polygon is a polygon that is both equilateral and equiangular. 

Theorem 58     The measure E of each exterior angle of an equilateral polygon of n sides is given by the formula:

And that was chapter 7! Good luck everyone and study hard! special shoutout thanks to Dobes for the ride home :)

Blessings, Em J