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Geometry A Common Core Curriculum

Ron Larson, Laurie Boswell

Chapter 6

Relationships Within Triangles

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Problem 1

COMPLETE THE SENTENCE Point $C$ is in the interior of $\angle \mathrm{DEF} .$ If $\angle \mathrm{DEC}$ and $\angle \mathrm{CEF}$ are congruent, then $\mathrm{EC}$ is the _____ of $\angle \mathrm{DEF}$.

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Problem 2

DIFFERENT WORDS, SAME QUESTIONS Which is different? Find both answers.

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Problem 3

In Exercises $3-6,$ Find the indicated measure. Explain your reasoning.
$$
\mathrm{GH}
$$

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Problem 4

In Exercises $3-6,$ Find the indicated measure. Explain your reasoning.
$$
\mathrm{QR}
$$

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Problem 5

In Exercises $3-6,$ Find the indicated measure. Explain your reasoning.
$$
A B
$$

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Problem 6

In Exercises $3-6,$ Find the indicated measure. Explain your reasoning.
$$
\mathrm{UW}
$$

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Problem 7

In Exercises $7-10,$ tell whether the information in the diagram allows you to conclude that point $\mathrm{P}$ lies on the perpendicular bisector of $\overline{\mathrm{LM}}$ . Explain your reasoning.

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Problem 8

In Exercises $7-10,$ tell whether the information in the diagram allows you to conclude that point $\mathrm{P}$ lies on the perpendicular bisector of $\overline{\mathrm{LM}}$ . Explain your reasoning.

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Problem 9

In Exercises $7-10,$ tell whether the information in the diagram allows you to conclude that point $\mathrm{P}$ lies on the perpendicular bisector of $\overline{\mathrm{LM}}$ . Explain your reasoning.

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Problem 10

In Exercises $7-10,$ tell whether the information in the diagram allows you to conclude that point $\mathrm{P}$ lies on the perpendicular bisector of $\overline{\mathrm{LM}}$ . Explain your reasoning.

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Problem 11

In Exercises $11-14$ , Find the indicated measure. Explain your reasoning.
$$
\mathrm{m} \angle \mathrm{ABD}
$$

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Problem 12

In Exercises $11-14$ , Find the indicated measure. Explain your reasoning.
$$
\mathrm{PS}
$$

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Problem 13

In Exercises $11-14$ , Find the indicated measure. Explain your reasoning.
$$
\mathrm{m} \angle \mathrm{K} \mathrm{JL}
$$

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Problem 14

In Exercises $11-14$ , Find the indicated measure. Explain your reasoning.
$$
\mathrm{FG}
$$

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Problem 15

In Exercises 15 and $16,$ tell whether the information in the diagram allows you to conclude that EH bisects $\angle$ FEG. Explain your reasoning. (See Example $4 . )$

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Problem 16

In Exercises 15 and $16,$ tell whether the information in the diagram allows you to conclude that EH bisects $\angle$ FEG. Explain your reasoning. (See Example $4 . )$

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Problem 17

In Exercises 17 and $18,$ tell whether the information in the diagram allows you to conclude that $ \mathrm{DB} $ $DC $. Explain your reasoning.

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Problem 18

In Exercises 17 and $18,$ tell whether the information in the diagram allows you to conclude that $ \mathrm{DB} $ $DC $. Explain your reasoning.

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Problem 19

In Exercises $19-22,$ write an equation of the perpendicular bisector of the segment with the given endpoints.
$$
\mathrm{M}(1,5), \mathrm{N}(7,-1)
$$

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Problem 20

In Exercises $19-22,$ write an equation of the perpendicular bisector of the segment with the given endpoints.
$$
Q(-2,0), R(6,12)
$$

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Problem 21

In Exercises $19-22,$ write an equation of the perpendicular bisector of the segment with the given endpoints.
$$
\mathrm{U}(-3,4), \mathrm{V}(9,8)
$$

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Problem 22

In Exercises $19-22,$ write an equation of the perpendicular bisector of the segment with the given endpoints.
$$
\mathrm{Y}(10,-7), \mathrm{Z}(-4,1)
$$

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Problem 23

ERROR ANALYSIS In Exercises 23 and $24,$ describe and correct the error in the student's reasoning.
Because AD $A E_{3}$ AB will pass through point $C$ .

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Problem 24

ERROR ANALYSIS In Exercises 23 and $24,$ describe and correct the error in the student's reasoning.
By the Angle Bisector Theorem (Theorem 6.3$)$ , x l 5 .

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Problem 25

MODELING MATHEMATICS In the photo, the road is perpendicular to the support beam and $\overline{\mathrm{AB}} \cong \overline{\mathrm{CB}}$ .Which theorem allows you to conclude that $\overline{\mathrm{AD}} \cong \overline{\mathrm{CD}} ?$

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Problem 26

MODELING WITH MATHEMATICS The diagram shows the position of the goalie and the puck during shows the position of the goalie and the puck during a hockey game. The goalie is at point G and the puck is at point $\mathrm{P}$ .
a. What should be the relationship between $\mathrm{PG}$ and $\angle \mathrm{APB}$ to give the goalie equal distances to travel on each side of $\mathrm{PG}$ ?
b. How does $\mathrm{m} \angle \mathrm{APB}$ change as the puck gets closer to the goal? Does this change make it easier or more difficult for the goalie to defend the goal? Explain your reasoning.

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Problem 27

CONSTRUCTION Use a compass and straightedge to construct a copy of $\overline{\mathrm{XY}}$ . Construct a perpendicular bisector and plot a point $\mathrm{Z}$ on the bisector so that the distance between point $\mathrm{Z}$ and $\overline{\mathrm{X} \mathrm{Y}}$ is 3 centimeters. Measure $\overline{\mathrm{XZ}}$ and $\overline{\mathrm{YZ}}$ Which theorem does this construction demonstrate?

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Problem 28

WRITING Explain how the Converse of the Perpendicular Bisector Theorem (Theorem 6.2$)$ is related to the construction of a perpendicular bisector.

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Problem 29

REASONING What is the value of $x$ in the diagram?
A. 13
B. 18
C. 33
D. not enough information

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Problem 30

REASONING Which point lies on the perpendicular bisector of the segment with endpoints $\mathrm{M}(7,5)$ and $\mathrm{N}(-1,5) ?$
A. $(2,0)$
B. $(3,9)$
C. $(4,1)$
D. $(1,3)$

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Problem 31

MAKING AN ARGUMENT Your friend says it is impossible for an angle bisector of a triangle to be the
same line as the perpendicular bisector of the opposite side. Is your friend correct? Explain your reasoning.

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Problem 32

PROVING A THEOREM Prove the Converse of the Perpendicular Bisector Theorem (Thm. 6.2$)$ .
(Hint: Construct a line through point $C$ perpendicular to $\overline{A B}$ at point $P . )$
Given $\mathrm{CA}=\mathrm{CB}$ Prove Point $\mathrm{Clies}$ on the perpendicular bisector of $\overline{\mathrm{AB}}$ .

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Problem 33

PROVING A THEOREM Use a congruence theorem to prove each theorem.
a. Angle Bisector Theorem (Thm. 6.3$)$
b. Converse of the Angle Bisector Theorem (Thm. 6.4 )

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Problem 34

HOW DO YOU SEE IT The Figure shows a map of a city. The city is arranged so each block north to
south is the same length and each block east to west is the same length.
a. Which school is approximately equidistant from both hospitals? Explain your reasoning.
b. Is the museum approximately equidistant from Wilson School and Roosevelt School? Explain
your reasoning.

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Problem 35

MATHEMATICAL CONNECTIONS Write an equation whose graph consists of all the points in the given
quadrants that are equidistant from the $\mathrm{x}$ - and y-axes.
a. I and III
b. II and IV
c. I and II

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Problem 36

THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In
spherical geometry, all points are on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible for two lines to be
perpendicular but not bisect each other? Explain your reasoning.

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Problem 37

PROOF Use the information in the diagram to prove that $\overline{\mathrm{AB}} \cong \overline{\mathrm{CB}}$ if and only if points $\mathrm{D}, \mathrm{E},$ and $\mathrm{B}$ are collinear.

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Problem 38

PROOF Prove the statements in parts (a)-(c).
Given Plane $P$ is a perpendicular bisector of $\overline{X Z}$ at point $Y .$
Prove
a. $\overline{\mathrm{XW}} \cong \overline{\mathrm{ZW}}$
b. $\overline{\mathrm{XV}} \cong \overline{\mathrm{ZV}}$
c. $\angle \mathrm{VXW} \cong \angle \mathrm{VZW}$

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Problem 39

Classify the triangle by its sides.

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Problem 40

Classify the triangle by its sides.

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Problem 41

Classify the triangle by its sides.

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Problem 42

Classify the triangle by its sides.

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Problem 43

Classify the triangle by its sides.

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Problem 44

Classify the triangle by its sides.

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