5-3 Triangles Warm Up Problem of the Day Lesson Presentation Pre-Algebra

5-3 Triangles Warm Up Solve each equation. 1. 62 + x + 37 = 180 Pre-Algebra 5-3 Triangles Warm Up Solve each equation. 1. 62 + x + 37 = 180 2. x + 90 + 11 = 180 3. x + x + 18 = 180 4. 180 = 2x + 72 + x x = 81 x = 79 x = 81 x = 36

Problem of the Day What is the one hundred fiftieth day of a non-leap year? May 30

Learn to find unknown angles in triangles.

Vocabulary Triangle Sum Theorem acute triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle

If you tear off two corners of a triangle and place them next to the third corner, the three angles seem to form a straight line.

Draw a triangle and extend one side Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown. The sides of the triangle are transversals to the parallel lines. The three angles in the triangle can be arranged to form a straight line or 180°.

An acute triangle has 3 acute angles An acute triangle has 3 acute angles. A right triangle has 1 right angle. An obtuse triangle has 1 obtuse angle.

Additional Example 1A: Finding Angles in Acute, Right and Obtuse Triangles Find p in the acute triangle. 73° + 44° + p = 180° 117° + p = 180° –117° –117° P = 63°

Additional Example 1B: Finding Angles in Acute, Right, and Obtuse Triangles Find c in the right triangle. 42° + 90° + c = 180° 132° + c = 180° –132° –132° c = 48°

Additional Example 1C: Finding Angles in Acute, Right, and Obtuse Triangles Find m in the obtuse triangle. 23° + 62° + m = 180° 85° + m = 180° –85° –85° m = 95°

Try This: Example 1A Find a in the acute triangle. 88° + 38° + a = 180° 38° 126° + a = 180° –126° –126° a = 54° a° 88°

Try This: Example 1B Find b in the right triangle. 38° 38° + 90° + b = 180° 128° + b = 180° –128° –128° b = 52° b°

Try This: Example 1C Find c in the obtuse triangle. 24° + 38° + c = 180° 38° 62° + c = 180° 24° c° –62° –62° c = 118°

An equilateral triangle has 3 congruent sides and 3 congruent angles An equilateral triangle has 3 congruent sides and 3 congruent angles. An isosceles triangle has at least 2 congruent sides and 2 congruent angles. A scalene triangle has no congruent sides and no congruent angles.

Additional Example 2A: Finding Angles in Equilateral, Isosceles, and Scalene Triangles Find angle measures in the equilateral triangle. 3b° = 180° Triangle Sum Theorem 3b° 180° 3 3 = Divide both sides by 3. b° = 60° All three angles measure 60°.

Additional Example 2B: Finding Angles in Equilateral, Isosceles, and Scalene Triangles Find angle measures in the isosceles triangle. 62° + t° + t° = 180° Triangle Sum Theorem 62° + 2t° = 180° Combine like terms. –62° –62° Subtract 62° from both sides. 2t° = 118° 2t° = 118° 2 2 Divide both sides by 2. t° = 59° The angles labeled t° measure 59°.

Find angle measures in the scalene triangle. Additional Example 2C: Finding Angles in Equilateral, Isosceles, and Scalene Triangles Find angle measures in the scalene triangle. 2x° + 3x° + 5x° = 180° Triangle Sum Theorem 10x° = 180° Combine like terms. 10 10 Divide both sides by 10. x = 18° The angle labeled 2x° measures 2(18°) = 36°, the angle labeled 3x° measures 3(18°) = 54°, and the angle labeled 5x° measures 5(18°) = 90°.

Try This: Example 2A Find angle measures in the isosceles triangle. 39° + t° + t° = 180° Triangle Sum Theorem 39° + 2t° = 180° Combine like terms. –39° –39° Subtract 39° from both sides. 2t° = 141° 2t° = 141° 2 2 Divide both sides by 2 39° t° = 70.5° t° The angles labeled t° measure 70.5°. t°

Find angle measures in the scalene triangle. Try This: Example 2B Find angle measures in the scalene triangle. 3x° + 7x° + 10x° = 180° Triangle Sum Theorem 20x° = 180° Combine like terms. 20 20 Divide both sides by 20. x = 9° 10x° The angle labeled 3x° measures 3(9°) = 27°, the angle labeled 7x° measures 7(9°) = 63°, and the angle labeled 10x° measures 10(9°) = 90°. 3x° 7x°

Try This: Example 2C Find angle measures in the equilateral triangle. 3x° = 180° Triangle Sum Theorem 3x° 180° 3 3 = x° x° = 60° x° x° All three angles measure 60°.

Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions The second angle in a triangle is six times as large as the first. The third angle is half as large as the second. Find the angle measures and draw a possible picture. Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle measure. 12

Additional Example 3 Continued Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle. 12 x° + 6x° + 3x° = 180° Triangle Sum Theorem 10x° = 180° Combine like terms. 10 10 Divide both sides by 10. x° = 18°

Additional Example 3 Continued Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle. 12 x° = 18° The angles measure 18°, 54°, and 108°. The triangle is an obtuse scalene triangle. 3 • 18° = 54° 6 • 18° = 108° X° = 18°

Try This: Example 3 The second angle in a triangle is three times larger than the first. The third angle is one third as large as the second. Find the angle measures and draw a possible picture. Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = x° = third angle measures. 13

Try This: Example 3 Continued Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = 3x° = third angle. 13 x° + 3x° + x° = 180° Triangle Sum Theorem 5x° = 180° Combine like terms. 5 5 Divide both sides by 5. x° = 36°

Try This: Example 3 Continued Let x° = the first angle measure. Then 3x° = second angle measure, and (3x°) = x° = third angle. 13 The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle. x° = 36° 3 • 36° = 108° x° = 36° 36° 108°

Lesson Quiz: Part 1 1. Find the missing angle measure in the acute triangle shown. 38° 2. Find the missing angle measure in the right triangle shown. 55°

Lesson Quiz: Part 2 3. Find the missing angle measure in an acute triangle with angle measures of 67° and 63°. 50° 4. Find the missing angle measure in an obtuse triangle with angle measures of 10° and 15°. 155°