Section 4.2Angles of Triangles

The Triangle Angle-Sum Theorem can be used to determine the measure of the third angle of a triangle when the other two angle measures are known.

Concept 2

Example 1: SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle.

Auxiliary line: an extra line or segment drawn in a figure to help analyze geometry relationships.

Find m1 first because the measure of two angles of the triangle are known. Use the Vertical Angles Theorem to find m2. Then you will have enough information to find the measure of 3.

m1 + 43° + 74° = 180° Triangle Angle-Sum Theorem

m1 + 117° = 180° Simplify

m1 = 63° Subtract 117 from each side.

1 and 2 are congruent vertical angles. So, m2 = 63°.

m3 + 63° + 79° = 180° Triangle Angle-Sum Theorem

m3 + 142° = 180° Simplify

m3 = 38° Subtract 142 from each side.

Therefore, m1 = 63°, m2 = 63°, and m3 = 38°.

Exterior Angle: an angle formed by one side of the triangle and the extension of an adjacent side. Each exterior angle of a triangle has two remote interior angles that are not adjacent to the exterior angle.

Ð4 is an exterior angle of ∆ABC. Its two remote interior angles are 1Ð and Ð3.

Concept 3

A flow proof uses statements written in boxes and arrows to show the logical progression of an argument. The reason justifying each statement is written below the box. See below for an example.

Example 2: GARDENING Find the measure of FLW in the fenced flower garden shown.

mLOW + mOWL = mFLW Exterior Angle Theorem

x + 32 = 2x – 48 Substitution

32 = x – 48 Subtract x from each side.

80 = x Add 48 to each side.

So, mFLW = 2(80) – 48 or 112°.

Concept 5

A corollary is a theorem with a proof that follows as a direct result of another theorem.

Example 3: Find the measure of each numbered angle.

m1 = 48° + 56° Exterior Angle Theorem

= 104° Simplify.

m1 + m2 = 180° If 2 s form a linear pair, theyare supplementary.

104° + m2 = 180° Substitution

m2 = 76° Subtract 104° from each side.

m4 = 48° Subtract 132° from each side.

132° + m4 = 180° Simplify.

56° + 76° + m4 = 180° Substitution

(90° – 34°) + m2 + m4 = 180° Triangle Angle-Sum Theorem

= 42° Simplify.

m3 = 90° – 48° If 2 s form a right angle, they are complementary.

m5 + 41° + 90° = 180° Triangle Angle-Sum Theorem

m5 + 131° = 180° Simplify

m5 = 49° Subtract 131° from each side.

m1 = 104°, m2 = 76°, m3 = 42°, m4 = 48°, m5 = 49°

Review & Practice!

1. Classify ΔRST as acute, equiangular, obtuse, or right.

2. Find y if ΔRST is an isosceles triangle with RS RT.___ ___

3. Find x if ΔABC is an equilateral triangle.

a) acute

b) scalene

c) isosceles

d) equiangular

4. Which is not a classification for ΔFGH?

5. Find a) m1.

b) m2.

c) m3.

d) m4.

e) m5.