5-3 Triangles

Learn to find unknown angles and identify possible side lengths in triangles.

5-3 Triangles

VocabularyTriangle Sum Theorem

acute triangle

right triangle

obtuse triangle

equilateral triangle

isosceles triangle

scalene triangle

Triangle Inequality Theorem

5-3 Triangles

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.

5-3 Triangles

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

5-3 Triangles

Additional Example 1A: Finding Angles in Acute, Right, and Obtuse Triangles

Find c° in the right triangle.

42° + 90° + c° = 180°

132° + c° = 180°

c° = 48°

–132° –132°

5-3 Triangles

Additional Example 1B: 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°

5-3 Triangles

Additional Example 1C: Finding Angles in Acute, Right and Obtuse Triangles

Find p° in the acute triangle.

73° + 44° + p° = 180°

117° + p° = 180°

p° = 63°

–117° –117°

5-3 Triangles

Find b in the right triangle.

38° + 90° + b° = 180°

128° + b° = 180°

b° = 52°

–128° –128°

38°

b°

Check It Out: Example 1A

5-3 Triangles

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.

5-3 Triangles

Additional Example 2A: Finding Angles in Equilateral, Isosceles, and Scalene Triangles

62° + t° + t° = 180°62° + 2t° = 180°

2t° = 118°

–62° –62°

Find the angle measures in the isosceles triangle.

2t° = 118°2 2

t° = 59°

Triangle Sum TheoremCombine like terms.Subtract 62° from both sides.

Divide both sides by 2.

The angles labeled t° measure 59°.

5-3 Triangles

Additional Example 2B: Finding Angles in Equilateral, Isosceles, and Scalene Triangles

2x° + 3x° + 5x° = 180°

10x° = 180°

x = 18°

10 10

Find the 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°.

5-3 Triangles

Additional Example 2C: Finding Angles in Equilateral, Isosceles, and Scalene Triangles

Find the angle measures in the equilateral triangle.

3b° = 180°

b° = 60°

3b° 180°3 3

=

Triangle Sum Theorem

All three angles measure 60°.

Divide both sides by 3.

5-3 Triangles

Check It Out: Example 2A

39° + t° + t° = 180°39° + 2t° = 180°

2t° = 141°

–39° –39°

Find the angle measures in the isosceles triangle.

2t° = 141°2 2

t° = 70.5°

Triangle Sum TheoremCombine like terms.Subtract 39° from both sides.

Divide both sides by 2

t°t°

39°

The angles labeled t° measure 70.5°.

5-3 Triangles

3x° + 7x° + 10x° = 180°

20x° = 180°

x = 9°

20 20

Find the angle measures in the scalene triangle.

Triangle Sum Theorem

Combine like terms.Divide both sides by 20.

3x° 7x°

10x°

Check It Out: Example 2B

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°.

5-3 Triangles

Find the angle measures in the equilateral triangle.

3x° = 180°

x° = 60°

3x° 180°3 3

=

Triangle Sum Theorem

All three angles measure 60°.

Check It Out: Example 2C

x° x°

x°

5-3 Triangles

Tell whether a triangle can have sides with the given lengths. Explain.

Find the sum of the lengths of each pair of sides and compare it to the third side.

Additional Example 4A: Using the Triangle Inequality Theorem

8 ft, 10 ft, 13 ft

8 + 10 > 13?

18 > 13

10 + 13 > 13?

23 > 13

8 + 13 > 10?

21 > 10

A triangle can have these side lengths. The sum of the lengths of any two sides is greater than the length of the third side.

5-3 Triangles

Tell whether a triangle can have sides with the given lengths. Explain.

Find the sum of the lengths of each pair of sides and compare it to the third side.

Additional Example 4B: Using the Triangle Inequality Theorem

2 m, 4 m, 6 m

2 + 4 > 6?

6 > 6

A triangle cannot have these side lengths. The sum of the lengths of two sides is not greater than the length of the third side.

5-3 Triangles

Tell whether a triangle can have sides with the given lengths. Explain.

Find the sum of the lengths of each pair of sides and compare it to the third side.

Check It Out: Example 4

17 m, 15 m, 33 m

17 + 15 > 33?

32 > 33

A triangle cannot have these side lengths. The sum of the lengths of two sides is not greater than the length of the third side.

5-3 Triangles

Lesson Quiz: Part I

1. Find the missing angle measure in the acute triangle shown.

2. Find the missing angle measure in the right triangle shown.

38°

55°

5-3 Triangles

Lesson Quiz: Part II

3. Find the missing angle measure in an acute triangle with angle measures of 67° and 63°.

4. Tell whether a triangle can have sides with lengths of 4 cm, 8 cm, and 12 cm.

50°

No; 4 + 8 is not greater than 12