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Angles of Triangles

Angles of Triangles. 3-4. Support Beams. Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. EXAMPLE 1. Classify triangles by sides and by angles. SOLUTION.

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Angles of Triangles

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  1. Angles of Triangles 3-4

  2. Support Beams Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. EXAMPLE 1 Classify triangles by sides and by 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.

  3. Classify PQOby its sides. Then determine if the triangle is a right triangle. Use the distance formula to find the side lengths. STEP1 2 2 – – ( ( ) ) OP = + + 2 2 – – ( ( ) ) y x x y y y x x 2 1 2 2 2 1 1 1 2 2 ( – ( ) (– 1 ) ) 0 2 – 0 2.2 + = = 5 OQ = 2 2 ( – ( ) 6 ) 0 – 0 3 6.7 + = = 45 EXAMPLE 2 Classify a triangle in a coordinate plane SOLUTION

  4. PQ = 2 2 ( – ) 6 (– 1 ) ) 3 – ( 2 7.1 + = = Check for right angles. STEP2 The slope ofOPis 2 – 0 3 – 0 1 . – 2. The slope ofOQis = = – 2 – 0 2 6 – 0 1 – 2 The product of the slopes is – 1 , = 2 – ( ) 2 + 2 – ( ) so OPOQand POQ is a right angle. y x x y 2 2 1 1 50 ANSWER Therefore, PQOis a right scalene triangle. EXAMPLE 2 Classify a triangle in a coordinate plane

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

  6. Triangle ABChas 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. Use the distance formula to find the side lengths. STEP1 2 2 – – ( ( ) ) AB = + + 2 2 – ( ( ) ) x x y y y x y x 1 1 2 2 1 2 1 2 2 2 ( – ( ) ( 3 ) ) 0 3 – 0 4.2 + = = 18 BC = 2 2 ( – ( ) –3 ) 3 – 3 3 20 + = = 400 for Examples 1 and 2 GUIDED PRACTICE SOLUTION

  7. AC = 2 2 ( – ) (–3 0 ) ) 3 – ( 0 4.2 + = = 18 Check for right angles. STEP2 3 – 0 3 – 0 The slope ofABis . 1. The slope ofACis = – 1 = – 3 – 0 3 – 0 1(– 1) The product of the slopes is – 1 , = 2 – ( ) + 2 – ( ) so ABACand BAC is a right angle. x y x y 2 1 2 1 Therefore, ABCis a right Isosceles triangle. ANSWER for Examples 1 and 2 GUIDED PRACTICE

  8. Findm∠ JKM. ALGEBRA STEP1 Write and solve an equation to find the value of x. (2x – 5)° = 70° + x° x = 75 STEP2 Substitute 75 for xin 2x–5 to find m∠ JKM. 2x–5 = 2 75 –5 = 145 ANSWER The measure of∠ JKMis145°. EXAMPLE 3 Find an angle measure SOLUTION Apply the Exterior Angle Theorem. Solve for x.

  9. 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. EXAMPLE 4 Find angle measures from a verbal description 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.

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

  11. Find the measure of 1 in the diagram shown. x= 25 STEP1 Write and solve an equation to find the value of x. (5x – 10)° = 40° + 3x° 2x = 50 for Examples 3 and 4 GUIDED PRACTICE SOLUTION Apply the Exterior Angle Theorem. Solve for x.

  12. STEP2 Substitute 25 for xin 5x–10 to find 1. 5x–10 = 115 5 25 –10 = 180 1 + (5x – 10)° = 180° 1 + 115° = 1 = 65° ANSWER So measure of∠ 1in the diagram is 65°. for Examples 3 and 4 GUIDED PRACTICE

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

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

  15. In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg? A 2x x Q B C for Examples 3 and 4 GUIDED PRACTICE 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.

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

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