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California Geometry

Edward Burger, David J. Chard,Earlene J. Hall

Chapter 5

Properties and Attributes of Triangles

Educators


Problem 1

Compare the given measures.
(FIGURE CANNOT COPY)
$A C$ and $X Z$

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Problem 2

Compare the given measures.
(FIGURE CANNOT COPY)
$\mathrm{m} \angle S R T$ and $\mathrm{m} \angle Q R T$

Mary B.
Numerade Educator

Problem 3

Compare the given measures.
(FIGURE CANNOT COPY)
$K L$ and $K N$

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Problem 4

Find the range of values for $x .$
(FIGURE CANNOT COPY)

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Problem 5

Find the range of values for $x .$
(FIGURE CANNOT COPY)

Mary B.
Numerade Educator

Problem 6

Find the range of values for $x .$
(FIGURE CANNOT COPY)

Mary B.
Numerade Educator

Problem 7

Health A therapist can take measurements to gauge the flexibility of a patient's elbow joint. In which position is the angle measure at the elbow joint greater? Explain.

Mary B.
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Problem 8

Write a two-column proof. Given: $\overline{F H}$ is a median of $\triangle D F G$
$$\mathrm{m} \angle D H F>\mathrm{m} \angle G H F$$Prove: $D F>G F$
(FIGURE CANNOT COPY)

Mary B.
Numerade Educator

Problem 9

Compare the given measures.
(FIGURE CANNOT COPY)
$$\mathrm{m} \angle D C A \text { and } \mathrm{m} \angle B C A$$

Mary B.
Numerade Educator

Problem 10

Compare the given measures.
(FIGURE CANNOT COPY)
$$\mathrm{m} \angle G H J \text { and } \mathrm{m} \angle K L M$$

Mary B.
Numerade Educator

Problem 11

Compare the given measures.
(FIGURE CANNOT COPY)
$T U$ and $S V$

Mary B.
Numerade Educator

Problem 12

Find the range of values for $z$
(FIGURE CANNOT COPY)

Mary B.
Numerade Educator

Problem 13

Find the range of values for $z$
(FIGURE CANNOT COPY)

Mary B.
Numerade Educator

Problem 14

Find the range of values for $z$
(FIGURE CANNOT COPY)

Mary B.
Numerade Educator

Problem 15

Industry The operator of a backhoe changes the distance between the cab and the bucket by changing the angle formed by the arms. In which position is the distance from the cab to the bucket greater? Explain.
(IMAGES CANNOT COPY)

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Numerade Educator

Problem 16

Write a two-column proof.
(FIGURE CANNOT COPY)
Given: $\overline{J K} \cong \overline{N M}, \overline{K P} \cong \overline{M Q}, J Q>N P$
Prove: $\mathrm{m} \angle K>\mathrm{m} \angle M$

Mary B.
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Problem 17

Critical Thinking $A B C$ is an isosceles triangle with base $\overline{B C} . X Y Z$ is an isosceles triangle with base $\overline{Y Z}$. Given that $\overline{A B} \cong \overline{X Y}$ and $\mathrm{m} \angle A=\mathrm{m} \angle X,$ compare $B C$ and $Y Z .$

Mary B.
Numerade Educator

Problem 18

Compare. Write $<,>,$ or $=$
$$\mathrm{m} \angle Q R P \quad \mathrm{m} \angle S R P$$

Mary B.
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Problem 19

Compare. Write $<,>,$ or $=$
$$\mathrm{m} \angle Q P R \quad \mathrm{m} \angle Q R P$$

Mary B.
Numerade Educator

Problem 20

Compare. Write $<,>,$ or $=$
$$\mathrm{m} \angle P R S \quad \mathrm{m} \angle R S P$$

Mary B.
Numerade Educator

Problem 21

Compare. Write $<,>,$ or $=$
$$\mathrm{m} \angle R S P \quad \mathrm{m} \angle R P S$$

Mary B.
Numerade Educator

Problem 22

Compare. Write $<,>,$ or $=$
$$\mathrm{m} \angle Q P R \quad \mathrm{m} \angle R P S$$

Mary B.
Numerade Educator

Problem 23

Compare. Write $<,>,$ or $=$
$$\mathrm{m} \angle P S R \quad \mathrm{m} \angle P Q R$$

Mary B.
Numerade Educator

Problem 24

Make a conclusion based on the Hinge Theorem or its converse. (Hint: Draw a sketch.)
In $\triangle A B C$ and $\triangle D E F, \overline{A B} \cong \overline{D E}, \overline{B C} \cong \overline{E F}, \mathrm{m} \angle B=59^{\circ},$ and $\mathrm{m} \angle E=47^{\circ}$

Mary B.
Numerade Educator

Problem 25

Make a conclusion based on the Hinge Theorem or its converse. (Hint: Draw a sketch.)
$\triangle R S T$ is isosceles with base $\overline{R T}$. The endpoints of $\overline{S V}$ are vertex $S$ and a point $V$ on $\overline{R T}$. $R V=4,$ and $T V=5$

Mary B.
Numerade Educator

Problem 26

Make a conclusion based on the Hinge Theorem or its converse. (Hint: Draw a sketch.)
In $\triangle G H J$ and $\triangle K L M, \overline{G H} \cong \overline{K L},$ and $\overline{G I} \cong \overline{K M}$ $\angle G$ is a right angle, and $\angle K$ is an acute angle.

Mary B.
Numerade Educator

Problem 27

Make a conclusion based on the Hinge Theorem or its converse. (Hint: Draw a sketch.)
In $\triangle X Y Z, \overline{X M}$ is the median to $\overline{Y Z}$, and $Y X>Z X$.

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Problem 28

Write About It The picture shows a door hinge in two different positions. Use the picture to explain why Theorem $5-6-1$ is called the Hinge Theorem.

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Problem 29

Write About It Compare the Hinge Theorem to the SAS Congruence Postulate. How are they alike? How are they different?

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Problem 30

This problem will prepare you for the Concept Connection on page 364 The solid lines in the figure show an airline's routes between four cities.
a. A traveler wants to fly from Jackson $U$ ) to Shelby $(S)$ but there is no direct flight between these cities. Given that $\mathrm{m} \angle N S J<\mathrm{m} \angle H S I,$ should the traveler first fly to Newton Springs ( $N$ ) or to Hollis ( $H$ ) if he wants to minimize the number of miles flown? Why?
b. The distance from Shelby ( $S$ ) to Jackson $U$ ) is 182 mi. What is the minimum number of miles the traveler will have to fly?

Mary B.
Numerade Educator

Problem 31

$\overline{M L}$ is a median of $\triangle J K L .$ Which inequality best describes the range of values for $x ?$
(FIGURE CANNOT COPY)
(A) $x>2$
(B) $x>10$
(C) $3<x<4 \frac{2}{3}$
(D) $3<x<10$

Mary B.
Numerade Educator

Problem 32

$\overline{D C}$ is a median of $\triangle A B C .$ Which of the following statements is true?
(F) $B C<A C$
(G) $B C>A C$
(H) $A D=D B$
(J) $\triangle D C=A B$

Mary B.
Numerade Educator

Problem 33

Short Response Two groups start hiking from the same camp. Group A hikes 6.5 miles due west and then hikes 4 miles in the direction N 35 $^{\circ}$ W. Group B hikes 6.5 miles due east and then hikes 4 miles in the direction $N 45^{\circ} \mathrm{E}$ At this point, which group is closer to the camp? Explain.
(IMAGE CANNOT COPY)

Mary B.
Numerade Educator

Problem 34

Multi-Step In $\triangle X Y Z, X Z=5 x+15, X Y=8 x-6$ and $\mathrm{m} \angle \mathrm{XVZ}>\mathrm{m} \angle \mathrm{XVY}$. Find the range of values for $x$
(FIGURE CANNOT COPY)

Mary B.
Numerade Educator

Problem 35

Use these steps to write a paragraph proof of the Hinge Theorem. Given: $\overline{A B} \cong \overline{D E}, \overline{B C} \cong \overline{E F}, \mathrm{m} \angle A B C>\mathrm{m} \angle D E F$ Prove: $A C>D F$
(FIGURE CANNOT COPY)
a. Locate $P$ outside $\triangle A B C$ so that $\angle A B P \cong \angle D E F$ and $\overline{B P} \cong \overline{E F} .$ Show that $\triangle A B P \cong \triangle D E F$ and
thus $\overline{A P} \cong \overline{D F}$
b. Locate $Q$ on $\overline{A C}$ so that $\overline{B Q}$ bisects $\angle P B C .$ Draw $\overline{Q P}$. Show that $\triangle B Q P \cong \triangle B Q C$ and thus $\overline{Q P} \cong \overline{Q C}$
c. Justify the statements $A Q+Q P>A P, A Q+Q C=A C, A Q+Q C>A P$ $A C>A P,$ and $A C>D F$

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Problem 36

Find the range and mode, if any, of each set of data.
$$2,5,1,0.5,0.75,2$$

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Problem 37

Find the range and mode, if any, of each set of data.
$$95,97,89,87,85,99$$

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Numerade Educator

Problem 38

Find the range and mode, if any, of each set of data.
$$5,5,7,9,4,4,8,7$$

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Problem 39

For the given information, show that $m \| n$ State any postulates or theorems used. (Lesson $3-3$ ) (FIGURE CANNOT COPY)
$$\mathrm{m} \angle 2=(3 x+21)^{\circ}, \mathrm{m} \angle 6=(7 x+1)^{\circ}, x=5$$

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Problem 40

For the given information, show that $m \| n$ State any postulates or theorems used. (Lesson $3-3$ ) (FIGURE CANNOT COPY)
$$\mathrm{m} \angle 4=(2 x+34)^{\circ}, \mathrm{m} \angle 7=(15 x+27)^{\circ}, x=7$$

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Problem 41

Find each measure. (Lesson $5-4$ )
$$D F$$

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Problem 42

Find each measure. (Lesson $5-4$ )
$$B C$$

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Problem 43

Find each measure. (Lesson $5-4$ )
$$\mathrm{m} \angle B F D$$

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Numerade Educator