Angles of Triangles 3-4

EXAMPLE 1 Classify triangles by sides and by angles Support Beams Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. SOLUTION The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 55° , 55° , and 70° . It is an acute isosceles triangle.

Classify a triangle in a coordinate plane EXAMPLE 2 Classify a triangle in a coordinate plane Classify PQO by its sides. Then determine if the triangle is a right triangle. SOLUTION STEP 1 Use the distance formula to find the side lengths. OP = y 2 – 1 ( ) x + = 2 – ( ) (– 1 ) + 5 2.2 OQ = y 2 – 1 ( ) x + 2 = – ( ) 6 + 3 45 6.7

Classify a triangle in a coordinate plane EXAMPLE 2 Classify a triangle in a coordinate plane PQ = y 2 – 1 ( ) x + 3 – 2 ( ) 6 + = (– 1 ) 50 7.1 STEP 2 Check for right angles. The slope of OP is 2 – 0 – 2 – 0 = – 2. The slope of OQ is 3 – 0 6 – 0 = 2 1 . 1 The product of the slopes is – 2 2 = – 1 , so OP OQ and POQ is a right angle. Therefore, PQO is a right scalene triangle. ANSWER

GUIDED PRACTICE for Examples 1 and 2 Draw an obtuse isosceles triangle and an acute scalene triangle. obtuse isosceles triangle B A C acute scalene triangle P Q R

Use the distance formula to find the side lengths. GUIDED PRACTICE for Examples 1 and 2 Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its sides. Then determine if it is a right triangle. SOLUTION STEP 1 Use the distance formula to find the side lengths. AB = y 2 – 1 ( ) x + = 3 – ( ) 2 ( 3 ) + = 18 4.2 BC = y 2 – 1 ( ) x + 2 = – 3 ( ) –3 3 + – = 400 20

The product of the slopes is 1(– 1) = – 1 , GUIDED PRACTICE for Examples 1 and 2 AC = y 2 – 1 ( ) x + = 3 – ( ) 2 0 ) (–3 + 18 4.2 STEP 2 Check for right angles. The slope of AB is 3 – 0 = 1. The slope of AC is 3 – 0 – 3 – 0 = . – 1 The product of the slopes is 1(– 1) = – 1 , so AB AC and BAC is a right angle. Therefore, ABC is a right Isosceles triangle. ANSWER

Write and solve an equation to find the value of x. EXAMPLE 3 Find an angle measure ALGEBRA Find m∠ JKM. SOLUTION STEP 1 Write and solve an equation to find the value of x. (2x – 5)° = 70° + x° Apply the Exterior Angle Theorem. x = 75 Solve for x. STEP 2 Substitute 75 for x in 2x – 5 to find m∠ JKM. 2x – 5 = 2 75 – 5 = 145 The measure of ∠ JKM is 145°. ANSWER

EXAMPLE 4 Find angle measures from a verbal description ARCHITECTURE The tiled staircase shown forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle. SOLUTION First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

Find angle measures from a verbal description EXAMPLE 4 Find angle measures from a verbal description Use the corollary to set up and solve an equation. x° + 2x° = 90° Corollary to the Triangle Sum Theorem x = 30 Solve for x. So, the measures of the acute angles are 30° and 2(30°) = 60° . ANSWER

Find the measure of 1 in the diagram shown. GUIDED PRACTICE for Examples 3 and 4 Find the measure of 1 in the diagram shown. SOLUTION STEP 1 Write and solve an equation to find the value of x. (5x – 10)° = 40° + 3x° Apply the Exterior Angle Theorem. 2x = 50 Solve for x. x= 25

GUIDED PRACTICE for Examples 3 and 4 STEP 2 Substitute 25 for x in 5x – 10 to find 1. 5x – 10 = 5 25 – 10 = 115 1 + (5x – 10)° = 180 1 + 115° = 180° 1 = 65° So measure of ∠ 1 in the diagram is 65°. ANSWER

GUIDED PRACTICE for Examples 3 and 4 x 2x 3x Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°. ° SOLUTION A + B + C = 180° x + 2x + 3x = 180° 6x = 180° x = 30° B = 2x = 2(30) = 60° C = 3x = 3(30) = 90°

Use the corollary to set up & solve an equation. GUIDED PRACTICE for Examples 3 and 4 Find the measures of the acute angles of the right triangle in the diagram shown. SOLUTION Use the corollary to set up & solve an equation. (x – 6)° + 2x° = 90° Corollary to the Triangle Sum Theorem 3x = 96 x = 32 Solve for x. Substitute 32 for x in equation x – 6 = 32 – 6 = 26°. So, the measure of acute angle 2(32) = 64° ANSWER

GUIDED PRACTICE for Examples 3 and 4 In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg? A B C Q 2x x SOLUTION First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

Use the corollary to set up and solve an equation. GUIDED PRACTICE for Examples 3 and 4 Use the corollary to set up and solve an equation. x° + 2x = 90° Corollary to the Triangle Sum Theorem x = 30 Solve for x. So the measures of the acute angles are 30° and 2(30°) = 60° ACD is linear pair to ACD. So 30° + ACD = 180°. Therefore = ACD = 150°. ANSWER