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Problem 4
Applying the Triangle Theorems
How can you apply
your skills from
Problem 3 here?
Look at the diagram.
Notice that you have a
triangle and information
about interior and
exterior angles.
Multiple Choice When radar tracks an object,
the reflection of signals off the ground can
result in clutter. Clutter causes the receiver
to confuse the real object with its reflection,
called a ghost. At the right, there is a radar
receiver at A, an airplane at B, and the
airplane’s ghost at D. What is the value of x?
30
70
50
80
m∠A + m∠B = m∠BCD
x + 30 = 80
RK
O
HO
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A
x!
Triangle Exterior Angle
Theorem
80!
D
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
1. Justify Mathematical Arguments (1)(G) Write a paragraph proof to
prove the Triangle Angle-Sum Theorem (Theorem 3-11). Begin by
drawing an auxiliary line through vertex T.
Given: △STU
U
Prove: m∠S + m∠T + m∠U = 180
Proof
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completing your homework,
go to PearsonTEXAS.com.
Find the value of each variable.
2.
30!
80!
3.
40!
70!
4.
30!
30!
x!
x! y!
y!
z!
c!
Find each missing angle measure.
5.
6.
60!
1
114
7.
128.5!
2
63!
Lesson 3-5 Parallel Lines and Triangles
C
Subtract 30 from
each side.
The value of x is 50. The correct answer is B.
ME
30!
Substitute.
x = 50
NLINE
B
13!
45!
3 4
47!
S
T
8. A ramp forms the angles shown at the right.
What are the values of a and b?
a!
72!
b!
9. Analyze Mathematical Relationships (1)(F) What
is the measure of each angle of a triangle with three congruent angles? Explain.
10. A beach chair has different settings that change
the angles formed by its parts. Suppose m∠2 = 71
and m∠3 = 43. Find m∠1.
Use the given information to find the unknown angle
measures in the triangle.
2
11. The ratio of the angle measures of the acute angles
in a right triangle is 1∶2.
1
3
12. The measure of one angle of a triangle is 40. The
measures of the other two angles are in a ratio of 3∶4.
13. The measure of one angle of a triangle is 108. The
measures of the other two angles are in a ratio of 1∶5.
14. Analyze Mathematical Relationships (1)(F) The
angle measures of △RST are represented by 2x,
x + 14, and x - 38. What are the angle measures of
△RST ?
15. Prove the following theorem: The acute angles of a right triangle are
complementary.
Proof
B
Given: △ABC with right angle C
Prove: ∠A and ∠B are complementary.
A
C
Find the values of the variables and the measures of the angles.
16.
17. C
Q
B
(2x # 4)"
(8 x ! 1)"
(2x ! 9)"
x"
P
18.
E
(4x # 7)"
R
e!
d!
32!
c!
H
55!
A
F
19.
B
x!
y!
b!
a!
G
54!
A
z ! 52!
D
C
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20. Prove the Triangle Exterior Angle Theorem (Theorem 3-12).
Proof
2
The measure of each exterior angle of a triangle equals the
sum of the measures of its two remote interior angles.
1 4
3
Given: ∠1 is an exterior angle of the triangle.
Prove: m∠1 = m∠2 + m∠3
21. Without using the Triangle Angle-Sum Theorem as a reason,
write a two-column proof to prove that the acute angles of a right
triangle are complementary.
D
A
Given: △ABC with right angle ACB
B E
C
Prove: ∠BAC and ∠ABC are complementary.
22. Explain Mathematical Ideas (1)(G) Two angles of a triangle measure 64 and 48.
What is the measure of the largest exterior angle of the triangle? Explain.
23. Analyze Mathematical Relationships (1)(F) A right triangle has exterior angles at
each of its acute angles with measures in the ratio 13∶14. Find the measures of the
two acute angles of the right triangle.
A
24. In the figure at the right, CD # AB and CD bisects ∠ACB.
Find m∠DBF .
D
25. If the remote interior angles of an exterior angle of a triangle
are congruent, what can you conclude about the bisector of
the exterior angle? Justify your answer.
F B
TEXAS Test Practice
26. The measure of one angle of a triangle is 115. The other two angles are congruent.
What is the measure of each of the congruent angles?
A. 32.5
B. 57.5
C. 65
D. 115
27. One statement in a proof is “∠1 and ∠2 are supplementary angles.” The next
statement is “m∠1 + m∠2 = 180.” Which is the best justification for the second
statement based on the first statement?
F. The sum of the measures of two right angles is 180.
G. Angles that form a linear pair are supplementary.
H. Definition of supplementary angles
J. The measure of a straight angle is 180.
28. △ABC has one obtuse angle, m∠A = 21, and ∠C is acute.
a. What is m∠B + m∠C? Explain.
b. What is the range of whole numbers for m∠C? Explain.
c. What is the range of whole numbers for m∠B? Explain.
116
Lesson 3-5 Parallel Lines and Triangles
( 3 x ! 2)"
(5x ! 20)" C
2. Select Tools to Solve Problems (1)(C) Consider the following conjecture.
If two triangles have the same perimeter, then the triangles are congruent.
a. Select a real object that you can use to test the conjecture. Explain your choice.
b. Is the conjecture true? If not, make a new conjecture based on your results.
Explain your reasoning.
3. Explain Mathematical Ideas (1)(G) At least how many triangle measurements
must you know in order to guarantee that all triangles built with those
measurements will be congruent? Explain your reasoning.
4. Given: IE ≅ GH, EF ≅ HF,
Proof
5. Given: WZ ≅ ZS ≅ SD ≅ DW
Proof
F is the midpoint of GI
Prove: △WZD ≅ △SDZ
Prove: △EFI ≅ △HFG
W
G
E
Z
F
D
H
I
S
What other information, if any, do you need to prove the two triangles
congruent by SAS? Explain.
6.
7.
G
L
T
U
T
N
W
R
M
Q
V
S
8. Evaluate Reasonableness (1)(B) You and a friend are cutting triangles out of
felt for an art project. You want all the triangles to be congruent. Your friend tells
you that each triangle should have two 5-in. sides and a 40° angle. If you follow
this rule, will all your felt triangles be congruent? Explain.
Can you prove the triangles congruent? If so, write the congruence statement and
name the postulate you would use. If not, write not enough information and tell
what other information you would need.
9. A
10.
G
N
R
T
Y
H
W
156
Lesson 4-2 Triangle Congruence by SSS and SAS
K
P
D
11.
J
E
T
S
F
V
12. Use Representations to Communicate Mathematical
Ideas (1)(E) Sierpinski’s triangle is a famous geometric pattern.
To draw Sierpinski’s triangle, start with a single triangle and connect
the midpoints of the sides to draw a smaller triangle. If you repeat
this pattern over and over, you will form a figure like the one shown.
This particular figure started with an isosceles triangle. Are the
triangles outlined in red congruent? Explain.
13. Create Representations to Communicate
Mathematical Ideas (1)(E) Use a straightedge to draw
any triangle JKL. Then construct △MNP ≅ △JKL using
the given postulate.
a. SSS
b. SAS
14. Analyze Mathematical Relationships (1)(F) Suppose GH ≅ JK , HI ≅ KL, and
∠I ≅ ∠L. Is △GHI congruent to △JKL? Explain.
15. Given: FG } KL, FG ≅ KL
Proof
Prove: △FGK ≅ △KLF
F
G
16. Given: AB # CM, AB # DB, CM ≅ DB,
Proof
M is the midpoint of AB.
Prove: △AMC ≅ △MBD
D
L
B
C
K
M
A
TEXAS Test Practice
Y
17. What additional information do you need to prove that
△VWY ≅ △VWZ by SAS?
A. YW ≅ ZW
C. ∠Y ≅ ∠Z
B. ∠WVY ≅ ∠WVZ
D. VZ ≅ VY
V
W
Z
18. The measures of two angles of a triangle are 43 and 38. What is
the measure of the third angle?
F. 9
G. 81
H. 99
J. 100
19. Which method would you use to find the inverse of a conditional statement?
A. Negate the hypothesis only.
C. Negate the conclusion only.
B. Switch the hypothesis and
the conclusion.
D. Negate both the hypothesis and
the conclusion.
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HO
ME
RK
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NLINE
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Determine whether the triangles must be congruent. If so, name the postulate or
theorem that justifies your answer. If not, explain.
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completing your homework,
go to PearsonTEXAS.com.
1.
T
2.
M
3.
W
V
U
P
S
4. Given: ∠FJG ≅ ∠HGJ, FG } JH
Proof
R
N
O
Prove: △FGJ ≅ △HJG
Y
5. Given: PQ # QS, RS # SQ,
Proof
T is the midpoint of PR
Prove: △PQT ≅ △RST
G
F
Z
R
Q
J
T
H
S
P
6. Evaluate Reasonableness (1)(B) While helping your
family clean out the attic, you find the piece of
paper shown at the right. The paper contains clues
to locate a time capsule buried in your backyard.
The maple tree is due east of the oak tree in your
backyard. Will the clues always lead you to the
correct spot? Explain.
7. Connect Mathematical Ideas (1)(F) Anita says
that you can rewrite any proof that uses the AAS
Theorem as a proof that uses the ASA Postulate.
Do you agree with Anita? Explain.
8. Justify Mathematical
Arguments (1)(G) Can you prove
that the triangles at the right are
congruent? Justify your answer.
9. Given: ∠N ≅ ∠P, MO ≅ QO
Prove: △MON ≅ △QOP
Proof
M
N
10. Given: ∠1 ≅ ∠2, and
DH bisects ∠BDF
Proof
Prove: △BDH ≅ △FDH
D
O
Q
1
P
B
H
2
F
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11. Given: AB } DC, AD } BC
Proof
A
B
Prove: △ABC ≅ △CDA
C
D
12. Create Representations to Communicate Mathematical Ideas (1)(E) Draw two
noncongruent triangles that have two pairs of congruent angles and one pair of
congruent sides.
13. Given AD } BC and AB } DC, name as many pairs of congruent
triangles as you can.
B
C
E
14. Create Representations to Communicate Mathematical Ideas (1)(E)
Use a straightedge to draw a triangle. Label it △JKL. Construct
△MNP ≅ △JKL so that the triangles are congruent by ASA.
A
D
15. Prove the Angle-Angle-Side Theorem (Theorem 4-2). Use the diagram next to it
on page 158.
16. In △RST at the right, RS = 5, RT = 9, and m∠T = 30. Show that
there is no SSA congruence rule by constructing △UVW with
UV = RS, UW = RT , and m∠W = m∠T , but with △UVW R △RST .
R
9
5
30!
S
TEXAS Test Practice
17. Suppose RT ≅ ND and ∠R ≅ ∠N. What additional information do you need to
prove that △RTJ ≅ △NDF by ASA?
A. ∠T ≅ ∠D
C. ∠J ≅ ∠D
B. ∠J ≅ ∠F
D. ∠T ≅ ∠F
18. You plan to make a 2 ft-by-3 ft rectangular poster of class trip photos. Each photo
is a 4 in.-by-6 in. rectangle. If the photos do not overlap, what is the greatest
number of photos you can fit on your poster?
F. 4
H. 32
G. 24
J. 36
19. Write the converse of the true conditional statement below. Then determine
whether the converse is true or false.
If you are less than 18 years old, then you are too young to vote in the United
States.
162
Lesson 4-3 Triangle Congruence by ASA and AAS
T
Problem 2
Proof Writing a Proof Using the HL Theorem
D
B
Given: BE bisects AD at C,
AB # BC, DE # EC, AB ≅ DE
How can you get
started?
Identify the hypotenuse
of each right triangle.
Prove that the
hypotenuses are
congruent.
C
BE bisects AD.
AC ≅ DC
Given
Def. of bisector
∠ABC and
∠DEC are
right ⦞.
AB ⊥ BC
DE ⊥ EC
Given
Def. of ⊥ lines
E
A
Prove: △ABC ≅ △DEC
△ ABC and △ DEC
are right .
△ABC ≅ △DEC
Def. of right triangle
HL Theorem
AB ≅ DE
NLINE
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Given
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
1. Justify Mathematical Arguments (1)(G) Copy the flow chart and
complete the proof.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
R
S
T
Given: PS ≅ PT , ∠PRS ≅ ∠PRT
Prove: △PRS ≅ △PRT
∠PRS and ∠PRT are ≅.
Given
P
∠PRS and ∠PRT
are right ⦞.
a.
∠PRS and ∠PRT
are supplementary.
⦞ that form a linear
pair are supplementary.
△PRS and △PRT
are right .
b.
PS ≅ PT
c.
PR ≅ PR
△PRS ≅ △PRT
e.
d.
2. Study Exercise 1. Can you prove that △PRS ≅ △PRT without using the HL
Theorem? Explain.
3. Explain Mathematical Ideas (1)(G) Complete the paragraph proof. B
D
Given: ∠A and ∠D are right angles, AB ≅ DE
Prove: △ABE ≅ △DEB
A
Proof: It is given that ∠A and ∠D are right angles. So, a. ? by the
definition of right triangles. b. ? , because of the Reflexive Property of
Congruence. It is also given that c. ? . So, △ABE ≅ △DEB by d. ? .
176
Lesson 4-6 Congruence in Right Triangles
E
4. Given: HV # GT , GH ≅ TV ,
I is the midpoint of HV
5. Given: PM ≅ RJ ,
PT # TJ , RM # TJ ,
M is the midpoint of TJ
Proof
Proof
Prove: △IGH ≅ △ITV
Prove: △PTM ≅ △RMJ
G
P
V
I
H
T
T
J
M
R
Connect Mathematical Ideas (1)(F) For what values of x and y are the triangles
congruent by HL?
6.
7.
x
x!3
3y
3y ! x
y!1
x!5
y"x
y!5
8. Apply Mathematics (1)(A) △ABC and △PQR are
right triangular sections of a fire escape, as shown.
Is each story of the building the same height?
Explain.
9. Connect Mathematical Ideas (1)(F) “Aha!”
exclaims your classmate. “There must be an HA
Theorem, sort of like the HL Theorem!” Is your
classmate correct? Explain.
10. Given: △LNP is isosceles with base NP,
Proof
MN # NL, QP # PL, ML ≅ QL
C
B
A
R
Prove: △MNL ≅ △QPL
L
M
Q
N
P
P
Q
Create Representations to Communicate Mathematical Ideas (1)(E)
Copy the triangle and construct a triangle congruent to it using the
given method.
11. SAS
12. HL
13. ASA
14. SSS
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15. Given: △GKE is isosceles with
base GE, ∠L and ∠D are
right angles, and K is the
midpoint of LD.
Proof
16. Given: LO bisects ∠MLN ,
OM # LM, ON # LN
Proof
Prove: △LMO ≅ △LNO
M
Prove: LG ≅ DE
L
K
O
D
L
G
N
E
17. Justify Mathematical Arguments (1)(G) Are the
triangles at the right congruent? Explain.
C
F
5
13
5
B
E
13
A
Analyze Mathematical Relationships (1)(F) For Exercises 18 and 19,
use the figure at the right.
D
B
18. Given: BE # EA, BE # EC, △ABC is equilateral
Proof
Prove: △AEB ≅ △CEB
E
A
19. Given: △AEB ≅ △CEB, BE # EA, BE # EC
C
Can you prove that △ABC is equilateral? Explain.
TEXAS Test Practice
20. You often walk your dog around the
neighborhood. Based on the diagram
at the right, which one of the following statements about distances is true?
A. SH = LH
C. SH 7 LH
B. PH = CH
D. PH 6 CH
School (S)
Park (P)
Home (H)
Café (C )
Library (L)
X
21. In equilateral △XYZ, name four pairs of congruent right
triangles. Explain why they are congruent.
P
Y
178
Lesson 4-6 Congruence in Right Triangles
S
R
Q
Z
Problem 4
TEKS Process Standard (1)(G)
Proof Separating Overlapping Triangles
C
Given: CA ≅ CE , BA ≅ DE
Prove: BX ≅ DX
NLINE
HO
ME
RK
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Which triangles are
useful here?
If △BXA ≅ △DXE,
then BX ≅ DX
because they are
corresponding parts. If
△BAE ≅ △DEA,
you will have enough
information to show
△BXA ≅ △DXE.
WO
B
D
X
E
A
B
D
X
A
B
E
D
A
Statements
E
E
A
Reasons
1) BA ≅ DE
1) Given
2) CA ≅ CE
2) Given
3) ∠CAE ≅ ∠CEA
3) Base ⦞ of an isosceles △ are ≅.
4) AE ≅ AE
4) Reflexive Property of ≅
5) △BAE ≅ △DEA
5) SAS
6) ∠ABE ≅ ∠EDA
s are ≅.
6) Corresp. parts of ≅ △
7) ∠BXA ≅ ∠DXE
7) Vertical angles are ≅.
8) △BXA ≅ △DXE
8) AAS
9) BX ≅ DX
s are ≅.
9) Corresp. parts of ≅ △
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
In each diagram, the red and blue triangles are congruent. Identify their common
side or angle.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. K
2.
P
L
E 3. X
D
T
N
W
G
F
M
Z
Y
Separate and redraw the indicated triangles. Identify any common sides or angles.
4. △PQS and △QPR
Q
P
5. △ACB and △PRB
A
K
P
T
B
L
O
C
R
S
6. △JKL and △MLK
J
M
R
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7. Justify Mathematical Arguments (1)(G) Complete the flow proof.
P
Given: ∠T ≅ ∠R, PQ ≅ PV
Prove: ∠PQT ≅ ∠PVR
Q
V
S
∠T ≅ ∠R
R
T
a.
∠TPQ ≅ ∠RPV
△TPQ ≅ △RPV
b.
∠PQT ≅ ∠PVR
e.
d.
PQ ≅ PV
c.
8. Given: RS ≅ UT , RT ≅ US
Proof
Prove: △RST ≅ △UTS
T
S
M
R
Prove: △QDA ≅ △UAD
U
Q
R
U
10. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4
Prove: △QET ≅ △QEU
11. Given: AD ≅ ED, D is the midpoint of BF
Proof
Prove: △ADC ≅ △EDG
T
Q
3
4
A
G
1
2
E
A
D
V
W
Proof
9. Given: QD ≅ UA, ∠QDA ≅ ∠UAD
Proof
B
U
F
B
D
E
C
12. Explain Mathematical Ideas (1)(G) In the diagram at the right,
∠V ≅ ∠S, VU ≅ ST, and PS ≅ QV. Which two triangles are
congruent by SAS? Explain.
W
V
13. Identify a pair of overlapping congruent triangles in the
diagram. Then use the given information to write a proof
to show that the triangles are congruent.
Given: AC ≅ BC, ∠A ≅ ∠B
Q
P
X
U
Lesson 4-7 Congruence in Overlapping Triangles
S
T
A
B
F
E
D
C
182
R
STEM
14. Apply Mathematics (1)(A) The figure at the right is
part of a clothing design pattern, and it has the
following relationships.
G
E
B
J
4 H 8 9
I
Ě GC # AC
Ě AB # BC
Ě AB } DE } FG
Ě m∠A = 50
A
D
1
F
2
7
5
3
6
C
Ě △DEC is isosceles with base DC.
a. Find the measures of all the numbered angles in the figure.
b. Suppose AB ≅ FC. Name two congruent triangles and explain how you can
prove them congruent.
15. Given: AC ≅ EC , CB ≅ CD
Proof
16. Given: QT # PR, QT bisects PR,
QT bisects ∠VQS
Q
Prove: VQ ≅ SQ
P
Proof
Prove: ∠A ≅ ∠E
C
B
A
D
F
V
E
R
S
T
17. Create Representations to Communicate Mathematical Ideas (1)(E) Draw a
AB } DC, AD } BC, and diagonals AC and DB intersecting
at E. Label your diagram to indicate the parallel sides.
Proof quadrilateral ABCD with
a. List all the pairs of congruent segments in your diagram.
b. Explain how you know that the segments you listed are congruent.
TEXAS Test Practice
18. According to the diagram at the right, which statement is true?
A. △DEH ≅ △GFH by AAS
C. △DEF ≅ △GFE by AAS
B. △DEH ≅ △GFH by SAS
D. △DEF ≅ △GFE by SAS
G
F
H
19. △ABC is isosceles with base AC. If m∠C = 37, what is m∠B?
F. 37
G. 74
H. 106
J. 143
E
20. Which word correctly completes the statement “All ? angles are
D
congruent”?
A. adjacent
B. supplementary C. right
D. corresponding
J
21. In the figure, LJ } GK and M is the midpoint of LG.
a. Copy the diagram. Then mark your diagram with the given information.
b. Prove △LJM ≅ △GKM.
M
L
c. Can you prove that △LJM ≅ △GKM another way? Explain.
G
K
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