© C

arne

gie

Lear

ning

Chapter 4 Assessments 1287

4

End of Chapter Test

Name Date

1. Triangles ABF and BCD share segment BF.B

DC

AF

E

a. If /BDC > /AFB and /BCD > /ABF, describe a sequence of steps that maps one triangle to the other.

b. If ___

EF is not parallel to ___

CD , what conclusion can you draw about points E and F?

2. Triangles ABC and ADE share angle A and ___

AB and ___

AC .

D

B

A C E8

10

15

69

12

a. Determine if triangle ABC is similar to ADE. Show your work.

b. If a segment through A that intersects BC at a point F and DE at a point G serves as the altitude of triangles ABC and ADE, how would you expect AF and AG to be related?

© C

arne

gie

Lear

ning

1288 Chapter 4 Assessments

4

3. In triangles ABD and ACE shown, BD || CE.

E

C

D

B

A

a. Use only the Triangle Proportionality Theorem and algebra to write a paragraph proof that if given BD || CE, then AB ___

AC 5 AD ___

AE .

b. The figure above is also used as part of a ski jump along a ski slope. The segment AC rests on the side of the slope and AE is the ramp. DB and EC are parallel support beams. The lengths of the ramp segments, AD and DE, are 15 feet and 10 feet respectively. If the length of the smaller support beam, BD is 12 feet, what is the length of the larger support beam, CE?

4. In the right triangle shown, the segment of length h is the altitude.

ba

4 8

h

a. Write a proportion that shows how all of these side lengths are related. Explain this relationship using the Right Triangle Altitude Theorem.

End of Chapter Test page 2

© C

arne

gie

Lear

ning

Chapter 4 Assessments 1289

4

b. Solve to find the values of a, b, and h.

5. The figure shown is constructed from four congruent right triangles of side lengths a and b, and hypotenuse c.

b c

a

a. Identify the shape of the entire figure. Justify your answer with a paragraph proof.

b. Identify the shape of the interior figure whose vertices are created by the intersections of the four triangles. Justify your answer with a paragraph proof.

c. Write area expressions for the entire figure as well as each of the parts of the figure.

End of Chapter Test page 3

Name Date

© C

arne

gie

Lear

ning

1290 Chapter 4 Assessments

4

d. Use the areas of these parts to write an equation, and then use it to prove the Pythagorean Theorem algebraically.

6. Alexandra holds a drawing on transparency paper XW in front of a flashlight held at point V. Her drawing is projected on the ceiling at ZY. She uses the measurements shown in inches to indirectly measure the length of the drawing projected on the ceiling.

W8

40

V

X

Z Y

a. If angle VWX is congruent to angle VYZ, how are the side lengths shown in the diagram related? Explain your reasoning.

b. If the length of XW, her drawing on the transparency paper, is 12 inches, what is the length of the drawing projected on the ceiling?

End of Chapter Test page 4