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Geometry Unit III Section 4.1: Triangle Sum Theorem The figure formed by 3 segments joining three noncollinear points A triangle is ___________________________________________________________________________ ___________________________________________________. C Triangles are classified according to their angles and sides.. Angle classification: 3 equal angles equiangular _____________________ 3 acute angles acute _____________________ 1 right angle right _____________________ 1 obtuse angle obtuse _____________________ A B Side classification: 3 equal sides equilateral _____________________ 2 equal sides isosceles _____________________ NO equal sides scalene _____________________ Examples: Classify each triangle according to its angles and sides. a) In ABC, m A 90 and AB AC . b) In DEF , m D m E m F 90 with no congruent sides. Right isosceles Acute scalene Theorem 4.1 Triangle Sum Theorem degrees (proof on p.196 of the text) The sum of the measures of the three angles of a triangle is 180 __________. Examples: Find the value of x. a) 35° 118° x° b) 71° 𝑥 + 35 + 118 = 180 𝑥 + 153 = 180 𝑥 = 27 c) 65° x° 180 − 71 − 65 = 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒 44 = 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒 180 − 44 = 𝑥 𝑥 = 136 (x–16)° x° 90 + 𝑥 + 𝑥 − 16 = 180 2𝑥 + 74 = 180 2𝑥 = 106 𝑥 = 53 The angle of x° in example b) is called an exterior angle of the triangle. An exterior angle of a triangle is formed by ___________________________________________________. Note that the exterior angle will Extending a side of the triangle Linear pair form a ________________________ with an interior angle of the triangle. 136 Note that __________________. 65 + 71 = 136 This work leads us to the In example b) we found x to equal _____. following theorem. Theorem 3.5.3 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to ________________ The sum of the measures of the 2 remote interior angles _____________________________________________________________. 2 1 m _____ m _____ 3 For the triangle to the right, m _____ 2 1 3 2𝑥 + 10 = 𝑥 + 65 Example: Find the value of x. 𝑥 + 10 = 65 x° 𝑥 = 55 65° (2x + 10)° A statement easily proven using a particular theorem A corollary is __________________________________________________________________________. Example c) on the previous page illustrates the following corollary: Corollary to the Triangle Sum Theorem Complementary or sum = 90 degrees The acute angles of a right triangle are __________________________.