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Warm Up

Apply Triangle Sum Properties. Warm Up. Lesson Presentation. Lesson Quiz. right. obtuse. ANSWER. ANSWER. ANSWER. acute. Warm-Up. Classify each angle as acute, obtuse, or right. 1. 90º. 2. 72º. 3. 116º. ~. 4. How do you know that 1 = 2?. 2. ANSWER.

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Warm Up

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  1. Apply Triangle Sum Properties Warm Up Lesson Presentation Lesson Quiz

  2. right obtuse ANSWER ANSWER ANSWER acute Warm-Up Classify each angle as acute, obtuse, or right. 1. 90º 2. 72º 3. 116º

  3. ~ 4. How do you know that 1 = 2? 2 ANSWER Alt. Int. s Thm. 1 Warm-Up

  4. Example 1 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.

  5. 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 x y y x 2 1 1 2 1 1 2 2 = 2.2 2 2 ( – ( ) (– 1 ) ) 0 2 – 0 + = 5 OQ = 2 2 ( – ( ) 6 ) 0 – 0 3 + = 45 = 6.7 Example 2 SOLUTION

  6. 2 2 PQ = ( – ) 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 y x x 2 1 2 1 50 Therefore, PQOis a right scalene triangle. ANSWER Example 2

  7. Draw an obtuse isosceles triangle and an acute scalene triangle. B SAMPLE ANSWER A C Q obtuse isosceles triangle R P acute scalene triangle Guided Practice

  8. 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. ANSWER isosceles; right triangle Guided Practice

  9. 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 = 75 –5 = 2 145 ANSWER The measure of∠ JKMis145°. Example 3 FindmJKM. SOLUTION Apply the Exterior Angle Theorem. Solve for x.

  10. 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. Example 4 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

  11. ANSWER x° + 2x° = 90° So, the measures of the acute angles are 30° and 2(30°) = 60°. x = 30 Example 4 Use the corollary to set up and solve an equation. Corollary to the Triangle Sum Theorem Solve forx.

  12. Find the measure of 1 in the diagram shown. The measure of∠ 1in the diagram is 65°. ANSWER Guided Practice

  13. Find the measure of each interior angle of ABC, where mA=x°, mB=2x°, and mC=3x°. Find the measures of the acute angles of the right triangle in the diagram shown. mA=30°, mB =60°, mC=90° ANSWER ANSWER 26° and 64° Guided Practice

  14. ANSWER 150° Guided Practice In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg?

  15. ANSWER isosceles; not a right angle Lesson Quiz 1. Graph ABC with vertices A(0, 6), B(– 4, – 1) and C(4, – 1). Classify it by its sides. Then determine if it is a right triangle.

  16. 2. Find x. Then classify the triangle by its angles. ANSWER ANSWER 22; acute 104° 3. Find the measure of the exterior angle shown. Lesson Quiz

  17. 4. Find x and y. ANSWER 82, 58 Lesson Quiz

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