1. 8-5 Triangles Warm Up 1. What are two angles whose sum is 90°? 2. What are two angles whose sum is 180°? 3. A part of a line between two points is called a _________. 4. Two lines that intersect at 90° are ______________. complementary angles supplementary angles segment perpendicular

2. 8-5 Triangles Learn to classify triangles and solve problems involving angle and side measures of triangles.

3. 8-5 Triangles Vocabulary acute triangle obtuse triangle right triangle congruent line segments scalene triangle isosceles triangle equilateral triangle

4. 8-5 Triangles A triangle is a closed figure with three line segments and three angles. Triangles can be classified by the measures of their angles. An acute triangle has only acute angles. An obtuse triangle has one obtuse angle. A right triangle has one right angle.

5. 8-5 Triangles To decide whether a triangle is acute, obtuse, or right, you need to know the measures of its angles. The sum of the measures of the angles in any triangle is 180°. You can see this if you tear the corners from a triangle and arrange them around a point on a line. By knowing the sum of the measures of the angles in a triangle, you can find unknown angle measures.

6. 8-5 Triangles Additional Example 1: Application D E F To classify the triangle, find the measure of D on the trophy. So the measure of D is 90°. Because DEF has one right angle, the trophy is a right triangle. Subtract the sum of the known angle measures from 180° m D = 180° – (38° + 52°) m D = 180° – 90° m D = 90° Sara designed this triangular trophy. The measure of E is 38°, and the measure of F is 52°. Classify the triangle.

7. 8-5 Triangles Check It Out: Example 1 To classify the triangle, find the measure of D on the trophy. So the measure of D is 136°. Because DEF has one obtuse angle, the trophy is an obtuse triangle. Subtract the sum of the known angle measures from 180° D E F m D = 180° – (22° + 22°) m D = 180° – 44° m D = 136° Sara designed this triangular trophy. The measure of E is 22°, and the measure of F is 22°. Classify the triangle.

8. 8-5 Triangles You can use what you know about vertical, adjacent, complementary, and supplementary angles to find the measures of missing angles.

9. 8-5 Triangles Additional Example 2A: Using Properties of Angles to Label Triangles Use the diagram to find the measure of each indicated angle. QTR and STR are supplementary angles, so the sum of mQTR and mSTR is 180°. mQTR = 180° – 68° = 112° P S R Q T 68° 55° QTR

10. 8-5 Triangles Additional Example 2B: Using Properties of Angles to Label Triangles  QRT and SRT are complementary angles, so the sum of mQRT and mSRT is 90°. m SRT = 180° – (68° + 55°) = 180° – 123° = 57° m QRT = 90° – 57° = 33° P S R Q T 68° 55°  QRT

11. 8-5 Triangles Check It Out: Example 2A Use the diagram to find the measure of each indicated angle. MNO and PNO are supplementary angles, so the sum of m  MNO and mPNO is 180°. m MNO = 180° – 44° = 136° L P O M N 44° 60° MNO

12. 8-5 Triangles Check It Out: Example 2B MON and PON are complementary angles, so the sum of m MON and m PON is 90°. m PON = 180° – (44° + 60°) = 180° – 104° = 76° m MON = 90° – 76° = 14° L P O M N 44° 60° MON

13. 8-5 Triangles Congruent line segments have the same length. Triangles can be classified by the lengths of their sides. A scalene triangle has no congruent sides. An isosceles triangle has at least two congruent sides. An equilateral triangle has three congruent sides.

14. 8-5 Triangles Additional Example 3: Classifying Triangles by Lengths of Sides Classify the triangle. The sum of the lengths of the sides is 19.5 in. c = 6.5 6.5 in. N L c M c + (6.5 + 6.5) = 19.5 c + 13 = 19.5 c + 13 – 13 = 19.5 – 13 6.5 in. Side c is 6.5 inches long. Because LMN has three congruent sides, it is an equilateral triangle.

15. 8-5 Triangles Check It Out: Example 3 Classify the triangle. The sum of the lengths of the sides is 15.6 in. d = 2.2 7.2 in. C A d B d + (7.2 + 7.2) = 16.6 d + 14.4 = 16.6 d + 14.4 – 14.4 = 16.6 – 14.4 7.2 in. Side d is 2.2 inches long. Because ABC has two congruent sides, it is an isosceles triangle.

16. 8-5 Triangles Lesson Quiz If the angles can form a triangle, classify the triangle as acute, obtuse, or right. 1. 37°, 53°, 90° 2. 65°, 110°, 25° 3. 61°, 78°, 41°4. 115°, 25°, 40° The lengths of three sides of a triangle are given. Classify the triangle. 5. 12, 16, 256. 10, 10, 15 not a triangle right acuteobtuse scalene isosceles