Holt CA Course 1

8-4 Triangles

Warm Up

California StandardsCalifornia Standards

Lesson Presentation

PreviewPreview

Holt CA Course 1

8-4 Triangles

Warm UpSolve each equation.1. 62 + x + 37 = 180

2. x + 90 + 11 = 180

3. 2x + 18 = 180

4. 180 = 3x + 72

x = 81

x = 79

x = 81

x = 36

Holt CA Course 1

8-4 Triangles

MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.Also covered: Review of 6MG2.2

California Standards

Holt CA Course 1

8-4 Triangles

Vocabulary

Triangle Sum Theorem acute triangle

right triangle obtuse triangle

equilateral triangle isosceles triangle

scalene triangle

Holt CA Course 1

8-4 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.

Holt CA Course 1

8-4 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. You can also show this in a drawing.

Holt CA Course 1

8-4 Triangles

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°.

Two sides of the triangle are transversals to the parallel lines.

Holt CA Course 1

8-4 Triangles

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

Holt CA Course 1

8-4 Triangles

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

A. Find p in the acute triangle.

73° + 44° + p° = 180°

117 + p = 180

p = 63

–117 –117

Triangle Sum Theorem

Subtract 117 from both sides.

Holt CA Course 1

8-4 Triangles

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

B. Find m in the obtuse triangle.

23° + 62° + m° = 180°

85 + m = 180

m = 95

–85 –85

Triangle Sum Theorem

Subtract 85 from both sides.

23

62

m

Holt CA Course 1

8-4 Triangles

Check It Out! Example 1

A. Find a in the acute triangle.

88° + 38° + a° = 180°

126 + a = 180

a = 54

–126 –126

88°

38°

a°

Triangle Sum Theorem

Subtract 126 from both sides.

Holt CA Course 1

8-4 Triangles

B. Find c in the obtuse triangle.

24° + 38° + c° = 180°

62 + c = 180

c = 118

–62 –62 c°

24°

38°

Check It Out! Example 1

Triangle Sum Theorem.

Subtract 62 from both sides.

Holt CA Course 1

8-4 TrianglesAdditional Example 2: Finding Angles in Equilateral,

Isosceles, and Scalene Triangles

62° + t° + t° = 180°62 + 2t = 180

2t = 118

–62 –62

A. Find the angle measures in the isosceles triangle.

2t = 1182 2

t = 59

Triangle Sum TheoremSimplify.Subtract 62 from both sides.

Divide both sides by 2.

The angles labeled t° measure 59°.

Holt CA Course 1

8-4 TrianglesAdditional Example 2: Finding Angles in Equilateral,

Isosceles, and Scalene Triangles

2x° + 3x° + 5x° = 180°

10x = 180

x = 18

10 10

B. Find the angle measures in the scalene triangle.

Triangle Sum Theorem

Simplify.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°.

Holt CA Course 1

8-4 TrianglesCheck It Out! Example 2

39° + t° + t° = 180°39 + 2t = 180

2t = 141

–39 –39

A. Find the angle measures in the isosceles triangle.

2t = 1412 2

t = 70.5

Triangle Sum TheoremSimplify.

Subtract 39 from both sides.

Divide both sides by 2

t°t°

39°

The angles labeled t° measure 70.5°.

Holt CA Course 1

8-4 Triangles

3x° + 7x° + 10x° = 180°

20x = 180

x = 9

20 20

B. Find the angle measures in the scalene triangle.

Triangle Sum Theorem

Simplify.Divide both sides by 20.

3x° 7x°

10x°

Check It Out! Example 2

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°.

Holt CA Course 1

8-4 Triangles

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 figure.

Let x° = the first angle measure. Then 6x° =

second angle measure, and (6x°) = 3x° =

third angle measure.

12

Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions

Holt CA Course 1

8-4 Triangles

Additional Example 3 Continued

x° + 6x° + 3x° = 180°

10x = 180 10 10

x = 18

Triangle Sum Theorem

Simplify.Divide both sides by 10.

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 figure.

Holt CA Course 1

8-4 Triangles

x° = 18°

6 • 18° = 108°

3 • 18° = 54°

The angles measure 18°, 108°, and 54°. The triangle is an obtuse scalene triangle.

Additional Example 3 Continued

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 figure.

Holt CA Course 1

8-4 Triangles

x° + 3x° + x° = 180°

5x = 180 5 5

x = 36

Triangle Sum Theorem

Simplify.Divide both sides by 5.

Check It Out! Example 3 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 figure.