Pre-Algebra 5.3 Triangles

Solve each equation x + 37 = x = x + x + 18 = = 2x x x = 81 x = 79 x = 81 x = 36 Warm Up

Learn to find unknown angles in triangles.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

3x° + 7x° + 10x° = 180° 20x° = 180° x = 9° Find angle measures in the scalene triangle. Triangle Sum Theorem Combine like terms. Divide both sides by 20. 3x°3x°7x°7x° 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°. Try This

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

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 Example: Finding Angles in a Triangle that Meets Given Conditions

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

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

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 Try This

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

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

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

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