1. Warm-Up Exercises Lesson 4.1, For use with pages 217-224 1.90º ANSWER right 2. 72º Classify each angle as acute, obtuse, or right. ANSWERacute

2. Warm-Up Exercises Lesson 4.1, For use with pages 217-224 3.116º ANSWER obtuse Classify each angle as acute, obtuse, or right. 4. How do you know that 1 = 2? ~ 2 1 ANSWER Alt. Int. s Thm.

3. Warm-Up Exercises EXAMPLE 1 Classifying Triangles

4. Warm-Up Exercises 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. Support Beams Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.

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

6. Warm-Up Exercises EXAMPLE 2 Classify a triangle in a coordinate plane PQ= y 2 –y 1 ( ) 2 x 2 –x 1 ( ) 2 + 3– 2( ) 2 6 ( ) 2 + – = (– 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

7. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 1. Draw an obtuse isosceles triangle and an acute scalene triangle. obtuse isosceles triangle B AC acute scalene triangle P Q R

8. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 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. ABC is a right Isosceles triangle. ANSWER

9. Warm-Up Exercises GUIDED PRACTICE Theorem

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

11. Warm-Up Exercises GUIDED PRACTICE Theorem

12. Warm-Up Exercises 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.

13. Warm-Up Exercises 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 x ° + 2x ° = 90 ° Solve for x. x = 30 So, the measures of the acute angles are 30 ° and 2(30 ° ) = 60 °. ANSWER

14. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 Find the measure of 1 in the diagram shown. 3. The measure of  1 in the diagram is 65 °. ANSWER

15. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 SOLUTION A + B + C = 180 ° x + 2x + 3x =180 ° 6x =180 ° x =30 ° B = 2x = 2(30) =60 ° C =3x = 3(30) =90 ° x 2x2x 3x3x 4. Find the measure of each interior angle of ABC, where m A = x, m B = 2x °, and m C = 3x °. °

16. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 5. Find the measures of the acute angles of the right triangle in the diagram shown. 26 ° and 64 ° ANSWER

17. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 6. 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 2x2x x ACD =150°. ANSWER

18. Warm-Up Exercises Homework 221: 1-7, 9-25 odd, 27-34, 42