Holt CA Course 1

8-4 Triangles

Vocabulary

Triangle Sum Theorem acute triangle

right triangle obtuse triangle

equilateral triangle isosceles triangle

scalene triangle midpoint

altitude

Holt CA Course 1

8-4 TrianglesA triangle is a three sided polygon.

What are the different types of triangles?

Equilateral Triangle: All Three Sides Are Congruent.Isosceles Triangle: Two Sides Are Of Equal Length.Scalene Triangle: No Sides Are Congruent Right Triangle: One Angle Is A Right Angle.Acute Triangle: All Angles Are Acute.Obtuse Triangle: One Angle is An Obtuse Angle.

What is the formula for finding the area of a triangle?

A = 1/2 bhb = base, h = height

Holt CA Course 1

8-4 Triangles

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 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 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

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.

Check It Out! Example 3

Let x° = the first angle measure. Then 3x° =

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

third angle measures.

13

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.

Holt CA Course 1

8-4 Triangles

x° = 36°

x° = 36°3 • 36° = 108°

The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle.

36° 36°

108°

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.

Holt CA Course 1

8-4 Triangles

The midpoint of a segment is the point that divides the segment into two congruent segments. An altitude of a triangle is a perpendicular segment from a vertex of the triangle to the line containing the opposite side.

Holt CA Course 1

8-4 Triangles

In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST.

Additional Example 3: Finding the Length of a Line Segment

20 ft

26 ft

S

U

V

T

Step 1 Find the length of TU.

__

__ ______

TU = UV1 2 T is the midpoint

of UV.

= (20) = 101 2

Holt CA Course 1

8-4 Triangles

In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST.

Additional Example 3 Continued

Step 2 Use the Pythagorean Theorem. Let ST = a and TU = b.

__ ______

Find the square root.a = 24

a2 + b2 = c2

a2 + 102 = 262

a2 + 100 = 676 –100 –100

a2 = 576

Pythagorean TheoremSubstitute 10 for b and 26 for c.

Simplify the powers. Subtract 100 from each side.

The length of ST is 24 ft, or ST is 24 ft.__

Holt CA Course 1

8-4 Triangles

In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB.

Check It Out! Example 3

Step 1 Find the length of BC.

__

______

__

BC = DC1 2 B is the midpoint

of DC.

= (14) = 71 2

14 in

25 in

A

C

D

B

Holt CA Course 1

8-4 TrianglesAdditional Example 3 Continued

Step 2 Use the Pythagorean Theorem. Let AB = a and BC = b.

__ ______

Find the square root.a = 24

a2 + b2 = c2

a2 + 72 = 252

a2 + 49 = 625 –49 –49

a2 = 576

Pythagorean TheoremSubstitute 7 for b and 25 for c.

Simplify the powers.

Subtract 49 from each side.

The length of AB is 24 in, or AB is 24 in.__

In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB.

Holt CA Course 1

8-4 TrianglesLesson 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°

Holt CA Course 1

8-4 TrianglesLesson Quiz: Part II

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°5. In the figure, M is the midpoint of AB and MD is t perpendicular to AB. Find the length of AB.

____

____

36 m

39 m

DM

A

B

30 m