8-5 Triangles Course 1 Warm Up Problem of the Day Lesson Presentation

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

Find the total number of shaded triangles in each figure. Problem of the Day Find the total number of shaded triangles in each figure. 3 6 10

Learn to classify triangles and solve problems involving angle and side measures of triangles.

Vocabulary acute triangle obtuse triangle right triangle scalene triangle isosceles triangle equilateral triangle

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. Acute triangle Obtuse triangle Right triangle

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.

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

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

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

Vertical angles are congruent Vertical angles are congruent. The sum of the measures of complementary angles is 90°. The sum of the measures of supplementary angles is 180°. Remember!

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

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

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

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

Triangles can be classified by the lengths of their sides 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.

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

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

Insert Lesson Title Here Course 1 8-5 Triangles Insert Lesson Title Here 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, 25 6. 10, 10, 15 not a triangle right acute obtuse scalene isosceles