Elementary Geometry for College Students

Daniel C. Alexander, Geralyn M. Koeberlein

Chapter 3

Triangles - all with Video Answers

Educators

Section 1

Congruent Triangles

Problem 1

In Exercises 1 to 4, consider the congruent triangles shown.
For the triangles shown, we can express their congruence with the statement $\triangle A B C \equiv \triangle F E D .$ By reordering the vertices, express this congruence with a different statement.
(GRAPH CANT COPY)

Phoebe Tyson

Phoebe Tyson

Numerade Educator

Problem 2

In Exercises 1 to 4, consider the congruent triangles shown.
With corresponding angles indicated, the triangles shown are congruent. Find values for $a, b,$ and $c$

Phoebe Tyson

Numerade Educator

Problem 3

In Exercises 1 to 4, consider the congruent triangles shown.
With corresponding angles indicated, find $\mathrm{m} \angle A$ if $\mathrm{m} \angle F=72^{\circ}$

Phoebe Tyson

Numerade Educator

Problem 4

In Exercises 1 to 4, consider the congruent triangles shown.
With corresponding angles indicated, find $\mathrm{m} \angle E$ if $\mathrm{m} \angle A=57^{\circ}$ and $\mathrm{m} \angle C=85^{\circ}$

Phoebe Tyson

Numerade Educator

Problem 5

Consider $\triangle A B C$ and $\triangle A B D$ in the figure shown. By the reason Identity, $\angle A \equiv \angle A$ and $\overline{A B} \equiv \overline{A B}$.
(a) If $\overline{B C} \cong \overline{B D}$ can you prove that $\triangle A B C \equiv \triangle A B D ?$
(b) If yes in part (a), by what reason are the triangles congruent?
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 5

In Exercises 13 to $18,$ use only the given information to state the reason why $\triangle A B C \cong \triangle D B C .$ Redraw the figure and use marks like those used in Exercises 9 to 12 .
$$\angle A \equiv \angle D, \overline{A C}=\overline{C D}, \text { and } \overrightarrow{C B} \text { bisects } \angle A C D$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 6

In a right triangle, the sides that form the right angle are the legs; the longest side (opposite the right angle) is the hypotenuse. Some textbooks say that when two right triangles have congruent pairs of legs, the right triangles are congruent by the reason LL. In our work, LL is just a special case of one of the postulates in this section. Which postulate is that?

James Kiss

Numerade Educator

Problem 7

In $\triangle A B C$, the midpoints of the sides are joined,
(a) What does intuition suggest regarding the relationship between $\triangle A E D$ and $\triangle F D E ?$ (We will prove this relationship later.)
(b) What does intuition suggest regarding $\triangle A E D$ and $\triangle E B F ?$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 8

(a) Suppose that you wish to prove that $\triangle R S T \equiv \triangle S R V$. Using the reason Identity, name one pair of corresponding parts that are congruent.
(b) Suppose you wish to prove that $\triangle R W T \cong \triangle S W V$. Considering the figure, name one pair of comesponding angles of these triangles that must be congruent.
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 9

In Exercises 9 to $12,$ congruent parts are indicated by like dashes (sides) or arcs (angles). State which method (SSS. SAS, ASA, or AAS) would be used to prove the nuo triangles congruent.
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 10

In Exercises 9 to $12,$ congruent parts are indicated by like dashes (sides) or arcs (angles). State which method (SSS. SAS, ASA, or AAS) would be used to prove the nuo triangles congruent.
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 11

In Exercises 9 to $12,$ congruent parts are indicated by like dashes (sides) or arcs (angles). State which method (SSS. SAS, ASA, or AAS) would be used to prove the nuo triangles congruent.
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 12

In Exercises 9 to $12,$ congruent parts are indicated by like dashes (sides) or arcs (angles). State which method (SSS. SAS, ASA, or AAS) would be used to prove the nuo triangles congruent.
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 13

In Exercises 13 to $18,$ use only the given information to state the reason why $\triangle A B C \cong \triangle D B C .$ Redraw the figure and use marks like those used in Exercises 9 to 12 .
$$\angle A \equiv \angle D, \overline{A B} \equiv \overline{B D}, \text { and } \angle 1 \equiv \angle 2$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 14

In Exercises 13 to $18,$ use only the given information to state the reason why $\triangle A B C \cong \triangle D B C .$ Redraw the figure and use marks like those used in Exercises 9 to 12 .
$$\angle A \equiv \angle D, \overline{A C} \equiv \overline{C D}, \text { and } B \text { is the midpoint of } \overline{A D}$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 15

In Exercises 13 to $18,$ use only the given information to state the reason why $\triangle A B C \cong \triangle D B C .$ Redraw the figure and use marks like those used in Exercises 9 to 12 .
$$\angle A \equiv \angle D, \overline{A C}=\overline{C D}, \text { and } \overrightarrow{C B} \text { bisects } \angle A C D$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 16

In Exercises 13 to $18,$ use only the given information to state the reason why $\triangle A B C \cong \triangle D B C .$ Redraw the figure and use marks like those used in Exercises 9 to 12 .
$$\angle A \equiv \angle D, \overline{A C} \equiv \overline{C D}, \text { and } \overline{A B} \equiv \overline{B D}$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 17

In Exercises 13 to $18,$ use only the given information to state the reason why $\triangle A B C \cong \triangle D B C .$ Redraw the figure and use marks like those used in Exercises 9 to 12 .
$$\overline{A C}=\overline{C D}, \overline{A B} \equiv \overline{B D}, \text { and } \overline{C B}=\overline{C B} \text { (by Identity) }$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 18

In Exercises 13 to $18,$ use only the given information to state the reason why $\triangle A B C \cong \triangle D B C .$ Redraw the figure and use marks like those used in Exercises 9 to 12 .
$$\angle 1 \text { and } \angle 2 \text { are right } \angle \mathrm{s}, \overline{A B} \equiv \overline{B D}, \text { and } \angle A \equiv \angle D$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 19

In Exercises 19 and 20 , the triangles to be proved congruent have been redrawn separately. Congruent parts are marked.
a) Name an additional pair of parts that are congruent by using the reason Identity.
b) Considering the congruent parts, state the reason why the triangles must be congruent.
$$\triangle A B C=\triangle A E D$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 20

In Exercises 19 and 20 , the triangles to be proved congruent have been redrawn separately. Congruent parts are marked.
a) Name an additional pair of parts that are congnuent by using the reason Identity.
b) Considering the congruent parts, state the reason why the triangles must be congruent.
$$\triangle M N P \equiv \triangle M Q P$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 21

In Exercises 21 to $24,$ the triangles named can be proven congnuent. Considering the congruent pairs marked, name the additional pair of parts that must be congnient in order to use the method named.

$$SAS$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 22

In Exercises 21 to $24,$ the triangles named can be proven congruent. Considering the congruent pairs marked, name the additional pair of parts that must be congruent in onder to use the method named.
$$ASA$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 23

In Exercises 21 to $24,$ the triangles named can be proven congruent. Considering the congruent pairs marked, name the additional pair of parts that must be congruent in onder to use the method named.
$$SSS$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 24

In Exercises 21 to $24,$ the triangles named can be proven congruent. Considering the congruent pairs marked, name the additional pair of parts that must be congruent in onder to use the method named.
$$AAS$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 25

In Exercises 25 and $26,$ complete each proof. Use the figure shown below.
Given: $\quad \overline{A B} \equiv \overline{C D}$ and $\overline{A D} \equiv \overline{C B}$
Prove:
$\triangle A B C \equiv \triangle C D A$
(TABLE CANT COPY)

James Kiss

Numerade Educator

Problem 26

Given:
$\overline{D C} \| \overline{A B}$ and $\overline{A D} \| \overline{B C}$
Prove:
$\triangle A B C \equiv \triangle C D A$
(TABLE CANT COPY)
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 26

In Exercises 27 to 32 , use SSS, SAS, ASA, or AAS to prove that the triangles are congruent.
$$\begin{aligned}
&\text {Given:}\\
&\overline{D C} \| \overline{A B} \text { and } \overline{A D} \| \overline{B C}\\
&\text { Prove: } \quad \triangle A B C \equiv \triangle C D A
\end{aligned}$$
(TABLE CANT COPY)

James Kiss

Numerade Educator

Problem 27

In Exercises 27 to $32,$ use SSS, SAS, ASA, or AAS to prove that the triangles are congruent.
$$\begin{aligned}
&\text {Given: } \quad \frac{\overrightarrow{P Q}}{M P} \equiv \frac{1}{N P}\\
&\text { Prove: } \quad \triangle M Q P=\triangle N Q P
\end{aligned}$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 28

In Exercises 27 to 32 , use SSS, SAS, ASA, or AAS to prove that the triangles are congruent.
Given:
$\overline{P Q} \perp \overline{M N}$ and
$\angle 1 \equiv \angle 2$
Prove: $\quad \triangle M Q P \equiv \Delta N Q P$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 29

In Exercises 27 to $32,$ use SSS, SAS, ASA, or AAS to prove that the triangles are congruent.
Given: $\quad \frac{\overline{A B}}{B C} \doteq \overline{B C}$ and $\overline{A B} \perp \overline{B D}$
Prove:
$\triangle A B C=\triangle A B D$

James Kiss

Numerade Educator

Problem 30

In Exercises 27 to $32,$ use SSS, SAS, ASA, or AAS to prove that the triangles are congruent.
Given:
$\overline{P N}$ bisects $\overline{M Q}$
$\angle M$ and $\angle Q$ are right angles Prove: $\quad \triangle P Q R=\triangle N M R$

James Kiss

Numerade Educator

Problem 31

In Exercises 27 to $32,$ use SSS, SAS, ASA, or AAS to prove that the triangles are congruent.
$$\begin{array}{ll}
\text {Given:} & \angle V R S \equiv \angle T S R \text { and } \overline{R V} \equiv \overline{T S} \\
\text {Prove:} & \triangle R S T \equiv \triangle S R V
\end{array}$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 32

In Exercises 27 to $32,$ use SSS, SAS, ASA, or AAS to prove that the triangles are congruent.
$$\begin{aligned}
&\text { Given: } \quad \overline{\boldsymbol{V S}}=\overline{\boldsymbol{T R}} \text { and } \angle T R S \equiv \angle V S R\\
&\text { Prove: } \quad \triangle R S T \equiv \Delta S R V
\end{aligned}$$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 33

In Exercises 33 to $36,$ the methods to be used are $5 \mathrm{ss}$, SAS. ASA, and AAS.
Given that $\triangle R S T \cong \triangle R V U,$ does it follow that $\triangle R S U$ is also congruent to $\triangle R V T$ ? Name the method, if any, used in arriving at this conclusion.
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 34

Given that $\angle S \equiv \angle V$ and $\overline{S T} \equiv \overline{U V},$ does it follow that $\triangle R S T \equiv \triangle R V U ?$ Which method, if any, did you use?

James Kiss

Numerade Educator

Problem 35

Given that $\angle A \equiv \angle E$ and $\angle B \equiv \angle D,$ does it follow that $\triangle A B C \equiv \triangle E D C T$ If so, cite the method used in arriving at this conclusion.
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 36

In Exercises 33 to $36,$ the methods to be used are $5 \mathrm{ss}$, SAS. ASA, and AAS.
Given that $\angle A \equiv \angle E$ and $\overline{B C} \equiv \overline{D C},$ does it follow that $\triangle A B C \equiv \triangle E D C 7$ Cite the method, if any, used in reaching this conclusion. (See the figure for Exercise 35.)

James Kiss

Numerade Educator

Problem 37

In quadrilateral $A B C D, \overline{A C}$ and $\overline{B D}$ are perpendicular bisectors of each other. Name all triangles that are congruent to:
a) $\triangle A B E$
b) $\triangle A B C$
c) $\triangle A B D$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 38

In $\triangle A B C$ and $\triangle D E F,$ you know that $\angle A \equiv \angle D$ $\angle C \equiv \angle F,$ and $\overline{A B} \equiv \overline{D E} .$ Before concluding that the triangles are congruent by ASA, you need to show that $\angle B \equiv \angle E .$ State the postulate or theorem that allows you to confirm this statement $(\angle B \cong \angle E)$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 39

In Exercises 39 and $40,$ complete each proof.
Given: $\quad$ Plane $M$
$C$ is the midpoint of $\overline{E B}$
$\overline{A D} \perp \overline{B E}$ and $\overline{A B} \| \overline{E D}$
Prove:
$\triangle A B C \equiv \triangle D E C$
(GRAPH CANT COPY)

James Kiss

Numerade Educator

Problem 40

In Exercises 39 and $40,$ complete each proof.
$$\begin{aligned}
&\text {Given: } \quad \overline{S P} \equiv \overline{S Q} \text { and } \overline{S T}=\overline{S V}\\
&\text { Prove: } \quad \triangle S P V \equiv \Delta S Q T \text { and } \triangle T P Q \equiv \Delta V Q P
\end{aligned}$$

Victoria Wieman

Victoria Wieman

Numerade Educator

Problem 41

Given:
$\angle A B C: \overline{R S}$ is the perpendicular bisector of $\overline{A B} ; \overline{R T}$ is the \begin{array}{l}
\text { perpendicular bisector } \\
\text { Prove: } \quad \frac{\text { of } B C}{A R} \equiv \overline{R C}
\end{array}
(GRAPH CANT COPY)

Nicholas Salas

Numerade Educator