Chapter 7: Polygons
7.1: Triangle Application Theorems
Theorem 50 The sum of the measures of the three angles of a triangle is 180o
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 = 180o. 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 = 180o Hence, m< A + m< B + m< C = 180o
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 180o, 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 Si of the measures of the angles of a polygon with n sides is given by the formula Si + (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 Se of the measures of the exterior angles of a polygon is given by the formula Se = 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
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