CONGRUENT TRIANGLES - Anoka-Hennepin School District 11

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CONGRUENT TRIANGLES

How does a cable-stayed

bridge work?

190

CHAPTER

4

APPLICATION: Bridges

On a cable-stayed bridge, the cables attached to the sides of each tower transfer the weight of the roadway to the tower.

You can see from the diagram below that the cables balance the weight of the roadway on both sides of each tower.

Not drawn to scale

B

E

A

D

C

G F

Think & Discuss

1. In the diagram above, what type of angle does each tower of the bridge make with the roadway?

2. Use the diagram above. Find at least one pair of acute angles that appear to be congruent and one pair of obtuse angles that appear to be congruent.

Learn More About It

You will prove that triangles formed by the cables and towers of a cable-stayed bridge are congruent in Exercise 16 on p. 234.

ERNET APPLICATION LINK Visit for more information about bridge construction.

INT

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191

Page 1 of 1

CHAPTER

4

PREVIEW

Study Guide

What's the chapter about?

Chapter 4 is about congruent triangles. Congruent triangles are triangles that are the same size and shape. In Chapter 4 you'll learn

? to prove triangles are congruent given information about their sides and angles. ? how to use congruent triangles to solve real-life problems.

KEY VOCABULARY

Review

? congruent segments, p. 19 ? acute angle, p. 28 ? right angle, p. 28 ? midpoint, p. 34 ? vertical angles, p. 44

? alternate interior angles,

p. 131

New

? isosceles triangle, p. 194 ? right triangle, p. 194 ? legs and hypotenuse of a

right triangle, p. 195

? interior angle, p. 196 ? exterior angle, p. 196 ? corollary, p. 197 ? congruent figures, p. 202 ? corresponding sides and

angles, p. 202

? coordinate proof, p. 243

PREPARE

STUDENT HELP

Study Tip "Student Help" boxes throughout the chapter give you study tips and tell you where to look for extra help in this book and on the Internet.

Are you ready for the chapter?

SKILL REVIEW Do these exercises to review key skills that you'll apply in this chapter. See the given reference page if there is something you don't understand.

xy USING ALGEBRA Solve the equation. (Skills Review, pp. 789 and 790)

1. 180 = 90 + x + 60 2. 6 = 2x + 2

3. 2x = 4x ? 6

4. 180 = 30 + 2x

5. 90 = 3x ? 90

6. 3x = 27 ? 6x

Use a protractor to draw an angle that has the given measure. Check your results by measuring the angle. (Review p. 27)

7. 30?

8. 135?

9. 72?

Use the diagram at the right. Write the theorem that supports each statement. (Review pp. 112 and 143)

10. TM1 ? TM2

11. TM3 ? TM4

12. TM1 ? TM5

1

35 2

4

STUDY STRATEGY

192 Chapter 4

Here's a study strategy!

Remembering Theorems

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theorems book.

that

you

? ?

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4.1 Triangles and Angles

What you should learn

GOAL 1 Classify triangles by their sides and angles, as applied in Example 2.

GOAL 2 Find angle measures in triangles.

Why you should learn it

To solve real-life problems, such as finding the measures of angles in a wing deflector in Exs. 45 and 46. AL LI

GOAL 1 CLASSIFYING TRIANGLES

A triangle is a figure formed by three segments joining three noncollinear points. A triangle can be classified by its sides and by its angles, as shown in the definitions below.

NAMES OF TRIANGLES

Classification by Sides

EQUILATERAL TRIANGLE

ISOSCELES TRIANGLE

SCALENE TRIANGLE

RE

FE

3 congruent sides

At least 2 congruent sides

No congruent sides

Classification by Angles

ACUTE TRIANGLE

EQUIANGULAR TRIANGLE

RIGHT TRIANGLE

OBTUSE TRIANGLE

A wing deflector is used to change the velocity of the water in a stream.

3 acute angles

3 congruent angles

1 right angle

Note: An equiangular triangle is also acute.

1 obtuse angle

194

E X A M P L E 1 Classifying Triangles

When you classify a triangle, you need to be as specific as possible.

a. ?ABC has three acute angles and no congruent sides. It is an acute scalene triangle. (?ABC is read as "triangle ABC.")

b. ?DEF has one obtuse angle and two congruent sides. It is an obtuse isosceles triangle.

A 65 B 58 57

C

D

130

F

E

Chapter 4 Congruent Triangles

Each of the three points joining the sides of a triangle is a vertex. (The plural of vertex is vertices.) For example, in ?ABC, points A, B, and C are vertices.

In a triangle, two sides sharing a common vertex are adjacent sides. In ?ABC, C?A and ? BA are adjacent sides. The third side, B?C, is the side opposite TMA.

side

C

opposite

TMA

B

adjacent sides

A

RIGHT AND ISOSCELES TRIANGLES The sides of right triangles and isosceles triangles have special names. In a right triangle, the sides that form the right angle are the legs of the right triangle. The side opposite the right angle is the hypotenuse of the triangle.

An isosceles triangle can have three congruent sides, in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third side is the base of the isosceles triangle.

hypotenuse

leg

leg base

leg

leg

Right triangle

Isosceles triangle

FOCUS ON APPLICATIONS

E X A M P L E 2 Identifying Parts of an Isosceles Right Triangle

The diagram shows a triangular loom.

a. Explain why ?ABC is an isosceles right triangle.

b. Identify the legs and the hypotenuse of ?ABC. Which side is the base of the triangle?

A

about 7 ft B

5 ft

5 ft

C

RE

FE

AL LI WEAVING

Most looms are used to weave rectangular cloth. The loom shown in the photo is used to weave triangular pieces of cloth. A piece of cloth woven on the loom can use about 550 yards of yarn.

SOLUTION

a. In the diagram, you are given that TMC is a right angle. By definition, ?ABC is a right triangle. Because AC = 5 ft and BC = 5 ft, ? AC ? B?C. By definition, ?ABC is also an isosceles triangle.

b. Sides ? AC and B?C are adjacent to the right angle, so they are the legs. Side ? AB is opposite the right angle, so it is the hypotenuse. Because ? AC ? B?C, side ? AB is also the base.

hypotenuse

and base

A

B

leg

leg

C

4.1 Triangles and Angles 195

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