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7.1 Notes Tuesday, December 02, 2008 1:45 PM Theorem 50: The sum of the measures of the three angles of a triangle is 180 Proof: Given: ABC Prove: mBAC + m B + mC = 180 A m C B Definition: An Exterior angle of a polygon is an angle that is adjacent to and supplementary to an interior angle of the polygon. Notes Page 1 ext Theorem 51: The measure of an exterior angle of a triangle is equal to the sum of the measure of the remote interior angles. M 1 = mA + mC A 1 C B Midline Theorem: A segment joining the midpoint of two sides of a triangle is parallel to the third side, and its length is onehalf the length of the third side. A D E C B Example 1: 80 Notes Page 2 80 100 z 55 60 y x Example 2: Find the mD A 80 D x x y y B C Notes Page 3 7.2 Notes Wednesday, December 03, 2008 10:45 AM No Choice Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. C F A B D AAS Theorem: We already use it :) Example 1: Given: A D Prove: E C D A B C E Statements 1. A D Reasons 1. Given Notes Page 4 E 2. ABE DBC 2. Verticals 's are 3. E C 3. No choice theorem Notes Page 5 7.3 notes Thursday, December 04, 2008 7:51 AM # of sides 3 4 5 6 7 8 9 10 12 15 n Polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon Pentadecagon n-gon Theorem 55: The sum Si of the measures of the angles of a polygon with n sides is given by the formula S i = (n - 2)180. B B C B C A A A C D D E Si= 180 Si= 360 Notes Page 6 Si= 540 Theorem 56: If one of the exterior angles is taken at each vertex, the sum SE of the measures of the exterior angles of a polygon is given by the formula S E = 360. 2 3 1 4 5 m1 + m2 + m3 + m4 + m5 = 360 Theorem 57: The number of diagonals that can be drawn in a polygon of n sides is given by the formula n(n 3) d 2 Example 1: Find the sum of the measures of the angles in a hexagon. Example 2: Find the number of diagonals that can be drawn in a heptagon. Notes Page 7 Example 3: What is the name of a polygon if the sum of the measures of its interior angles is 1080? Notes Page 8 7.4 Notes Thursday, December 04, 2008 12:58 PM Definition: A regular polygon is a polygon that is both equilateral and equiangular. equilateral triangle square Regular Pentagon Theorem 58: The measure of E of each exterior angle of an equiangular polygon of n sides is given by the formula E 360 n Example 1: How many degrees are there in each exterior angle of a regular pentagon? Example 2: How many degrees are there in each interior angle of a regular pentagon? Notes Page 9 pentagon? Example 3: If each exterior angle of a polygon is 18, how many sides does the polygon have? Notes Page 10