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8.
_
_
Given: AB is parallel to CD, ∠ACB ≅ ∠CAD.
A
B
Prove: △ABC ≅ △CDA
D
1. Given
2. ∠BAC ≅ ∠DCA
2. Alternate Interior Angles Theorem
¯ ≅ AC
¯
3. AC
9.
C
Statements
_
¯ is parallel to CD.
1. AB
Reasons
3. Reflexive Property of Congruence
4. ∠ACB ≅ ∠CAD
4. Given
5. △ABC ≅ △CDA
5. ASA Triangle Congruence Theorem
_
Given: _
∠H ≅ ∠J, G is the midpoint
_ of HJ,
FG is perpendicular to HJ.
F
Prove: △FGH ≅ △FGJ
H
Statements
© Houghton Mifflin Harcourt Publishing Company
1. ∠H ≅ ∠J
Reasons
1. Given
¯.
2. G is the midpoint of HJ
2. Given
¯ ≅ JG
¯
3. HG
3. Definition of midpoint
¯ is perpendicular to HJ
¯.
4. FG
4. Given
5. ∠FGH and ∠FGJ are right angles.
5. Definition of perpendicular
6. ∠FGH ≅ ∠FGJ
6. All right angles are congruent.
7. △FGH ≅ △FGJ
7. ASA Triangle Congruence Theorem
10. The figure shows quadrilateral PQRS. What additional information do
you need in order to conclude that △SPR ≅ △QRP by the ASA Triangle
Congruence Theorem? Explain.
∠SRP ≅ ∠QPR; you need to have two pairs of congruent
¯ as the included side.
corresponding angles with PR
a. Describe a sequence of two rigid motions that maps △LMN to △WXY.
‹ ›
−
¯ to WX
¯, then reflect △LMN across WX
Translate LM
.
b. How can you be sure that point N maps to point Y ?
Since ∠L ≅ ∠W and ∠M ≅ ∠X, the images of ¯
LN and ¯
MN lie
→
→
‾ , respectively. The image of N must lie on both
‾ and XY
on WY
rays, so the image is the intersection point Y.
GE_MNLESE385795_U2M05L2 239
239
Lesson 5.2
P
239
Q
S
_
_
11. Communicate Mathematical Ideas In the figure, WX is parallel to LM.
Module 5
J
G
R
Y
X
W
L
M
N
Lesson 2
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Write each proof.
B
_
15. Given: ∠A ≅ ∠E, C is the midpoint of AE.
¯ ≅ ED
¯
Prove: AB
C
A
E
D
Statements
1. ∠A ≅ ∠E
Reasons
1. Given
¯.
2. C is the midpoint of AE
2. Given
¯ ≅ EC
¯
3. AC
3. Definition of midpoint
4. ∠ACB ≅ ∠ECD
4. Vertical angles are congruent.
5. △ACB ≅ △ECD
5. ASA Triangle Congruence Theorem
¯ ≅ ED
¯
6. AB
6. CPCTC
16. The figure shows △GHJ and △PQR on a coordinate plane.
4
a. Explain why the triangles are congruent using the ASA Triangle
Congruence Theorem.
∠J ≅ ∠R, ∠H ≅ ∠Q (both are right angles), and
¯ (both are 4 units long). Two pairs of angles and
¯ ≅ QR
HJ
© Houghton Mifflin Harcourt Publishing Company
their included sides are congruent, so the triangles are
congruent by the ASA Triangle Congruence Theorem.
-4
Q
0
-2
P
J
x
R
4
-4
Possible answer: You can map △GHJ onto △PQR by a
translation two units left followed by a reflection across
the x-axis. By the definition of congruence in terms of rigid
motions, the triangles must be congruent.
GE_MNLESE385795_U2M05L2 241
Lesson 5.2
2
H
b. Explain why the triangles are congruent using rigid motions.
Module 5
241
y
G
241
Lesson 2
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%0/05&%*5$IBOHFTNVTUCFNBEFUISPVHI'JMFJOGP
$PSSFDUJPO,FZ/-"$""
CRITICAL THINKING
8.
_ _
Given that polygon ABCDEF is a regular hexagon, prove that AC ≅ AE .
C
B
Draw non-collinear points M, O, and U on the
board, connecting them to form an obtuse angle
with vertex O. Ask students to visualize a translation,
a rotation, and a reflection of the figure shown.
In each case, have them describe the effect on the
segment that connects point M with point U.
Sample response: The segment connecting points M
and U will follow the same movements as the rest of
the figure. Its length will remain the same no matter
what rigid-motion transformation is used.
D
A
F
E
Statements
9.
Reasons
1. ABCDEF is a regular hexagon.
1. Given
2. AB = AF and BC = FE
_ _
_ _
3. AB ≅ AF and BC ≅ FE
2. Definition of regular polygon
4. m∠B = m∠F
4. Definition of regular polygon
3. Definition of congruence in terms of rigid motion
5. ∠B ≅ ∠F
5. Definition of congruence in terms of rigid motion
6. △ABC ≅ △AFE
_ _
7. AC ≅ AE
6. SAS Triangle Congruence Theorem
7. CPCTC
A product designer is designing an easel
braces_
as
_ with
_ extra_
shown in the diagram.
Prove
_ that if BD ≅ FD and CD ≅ ED,
_
then the braces BE and FC are also congruent.
ª)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH$PNQBOZt*NBHF$SFEJUT
ª"OESFZVVJ4UPDL1IPUPDPN
D
C
A
B
GE_MNLESE385795_U2M05L3 251
Lesson 5.3
F
_ _
_ _
You are given that BD ≅ FD and CD ≅ ED. You also know that ∠D ≅ ∠D
by the reflexive property. Two sides and the included angle of △BDE are
congruent to two sides and the included angle of △FDC. The triangles are
congruent by the SAS Triangle Congruence Theorem. So, by CPCTC, the
¯ and are also congruent.
¯ and FC
braces BE
Module 5
251
E
251
Lesson 3
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10. An artist is framing a large picture and wants to put metal poles across the back to
strengthen the frame as shown in the diagram.
metal poles
_If the
_
_ are
_both the same
length and they bisect each other, prove that AB ≅ CD and AD ≅ CB.
A
AVOID COMMON ERRORS
Students may choose the wrong angle when SAS is
used to prove triangles congruent. Explain that the
angle must be formed by the sides. The included
angle is named by the letter the segments share.
B
E
D
C
_
_
Because
BD and_
AC bisect
_ each other, AE = CE and BE = DE,
_ _
so AE ≅ CE and BE ≅ DE by the definition of congruence. By the Vertical
Angle Theorem, you also know that ∠AEB ≅ ∠CED. Two sides and the
included angle of △AEB are congruent to two sides and the included
angle of △CED. The triangles are congruent
_ _ by the SAS Triangle
Congruence Theorem.
_ _ By CPCTC, AB ≅ CD. You can use similar reasoning
to show that AD ≅ CB
11. The figure shows a side panel of a skateboard ramp. Kalim wants to confirm that the
right triangles in the panel are congruent.
C
A
D
B
a. What measurements should Kalim take if he wants to confirm that the triangles
are congruent by SAS? Explain.
_
_
Measure AB and DB; so _
he can_
confirm
that two pairs of sides and their included
_ _
angles are congruent. (AB ≅ DB, CB ≅ CB, and ∠ABC ≅ ∠DBC)
Measure ∠ACB and ∠DCB; so he can confirm that two pairs
and their included
_of angles
_
sides are congruent. (∠ACB ≅ ∠DCB, ∠ABC ≅ ∠DBC, and CB ≅ CB)
Module 5
GE_MNLESE385795_U2M05L3.indd 252
252
© Houghton Mifflin Harcourt Publishing Company
b. What measurements should Kalim take if he wants to confirm that the triangles
are congruent by ASA? Explain.
Lesson 3
6/9/15 3:08 AM
SAS Triangle Congruence
252
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INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2"TLTUVEFOUTUPDPOTJEFSUIFDIBOHFTJO
H.O.T. Focus on Higher Order Thinking
26. Explain the Error Ava wants to know the distance JK across
a pond. She locates points as shown. She says that the distance
across the pond must be 160 ft by the SSS Triangle Congruence
Theorem. Explain her error.
L
210 ft
N
160 ft
190 ft
M
The distance is 160 ft, but the justification should be
190 ft
Given:
∠BFC ≅ ∠ECF, ∠BCF ≅ ∠EFC
_ _ _
_
AB ≅ DE, AF ≅ DC
Prove:
△ABF ≅ △DEC
K
A
B
1. ∠BFC ≅ ∠ECF, ∠BCF ≅ ∠EFC
―
C
F
Statements
―
E
1. Given
2. Reflexive Property of Congruence
3. △BFC ≅ △ECF
3. ASA Triangle Congruence Theorem
4. FB ≅ CE
4. CPCTC
―
―
――
―
D
Reasons
2. FC ≅ FC
―
Mike’s bench if it were to transform from the original
shape to complete collapse. Which properties of the
bench would change and how would they change?
Which properties would not change? Sample
answer: Change: During collapse, the area inside the
parallelogram would decrease continually; the
measures of the angles would change, with the
measures of one pair of opposite angles increasing
continually and the measures of the other pair
decreasing continually.
210 ft
the SAS Triangle Congruence Theorem.
27. Analyze Relationships Write a proof.
J
5. AB ≅ DE, AF ≅ DC
5. Given
6. △ABF ≅ △DEC
6. SSS Triangle Congruence Theorem
Not change: The lengths of the sides and the
perimeter of the parallelogram would not change.
AVOID COMMON ERRORS
Students may argue that Mike’s bench may be solidly
constructed with screws, nails, and glue, so that it
would not collapse. Stress that the bench is a
real-world object introduced here to model an
abstract geometrical principle, and that it is not a
perfect model. The important conclusion to draw is
the one relating to quadrilaterals, not to benches.
Lesson Performance Task
Mike
Michelle
© Houghton Mifflin Harcourt Publishing Company
Mike and Michelle each hope to get a contract with the city to build benches for commuters to sit on while waiting for
buses. The benches must be stable so that they don’t collapse, and they must be attractive. Their designs are shown.
Judge the two benches on stability and attractiveness. Explain your reasoning.
Answers will vary. While Mike's bench will probably be chosen as the more
attractive, students should note that Michelle's is more stable. There are many
quadrilaterals with the same side lengths as Mike's bench. But, by the SSS
Congruence Theorem, the triangles in Michelle's design cannot change shape.
Module 5
266
Lesson 4
EXTENSION ACTIVITY
GE_MNLESE385795_U2M05L4.indd 266
Have students construct models of Mike and Michelle’s benches from sturdy
pieces of cardboard or photo-print paper cut into long, narrow strips and attached
at the corners with brads. Students can explore further by making other polygons,
checking their stability, and explaining why, for example, there is no SSSSS
Pentagon Congruence Theorem. A 5-sided polygon is not stable. This means
that there are many different shapes that can be constructed from the same
5 sides of a pentagon.
6/9/15 2:29 AM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
SSS Triangle Congruence
266
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Reflect
EXPLAIN 2
3.
Using Angle-Angle-Side Congruence
Discussion The Third Angles Theorem says “If two angles of one triangle
are congruent to two angles of another triangle, then the third pair of
angles are congruent.” How could using this theorem simplify the proof of the
AAS Congruence Theorem?
Steps 2–8 could be simplified to one step, ∠B ≅ ∠E, making the whole
proof only three steps.
QUESTIONING STRATEGIES
4.
How does the AAS Congruence Theorem
differ from the ASA Congruence
Theorem? While both the ASA and AAS Congruence
Theorems require you to know that two pairs of
corresponding angles are congruent between two
triangles, the ASA Congruence Theorem requires
you to also know that a pair of corresponding
included sides are congruent. The AAS Congruence
Theorem requires you to know also that a pair of
corresponding non-included sides are congruent.
Could the AAS Congruence Theorem be used in the proof? Explain.
No, that is the theorem being proved, so it cannot be used until it has
been proven.
Explain 2
Example 2
Using Angle-Angle-Side Congruence
Use the AAS Theorem to prove the given triangles are congruent.
_
_
Given: AC ≅ EC and m‖n
D
E
Prove: △ABC ≅ △EDC
m
C
A
B
n
AC ≅ EC
If AAS is used as the method of proof, can the
triangles also be proved congruent using
ASA? Yes; you can use the Third Angles Theorem to
show the third pair of angles is congruent, and then
you can use ASA.
Given
mǁn
Given
∠E ≅ ∠A
△ABC ≅ △EDC
Alt. Int ! Thm.
AAS Cong. Thm.
∠B ≅ ∠D
© Houghton Mifflin Harcourt Publishing Company
Alt. Int ! Thm.
_ _ _ _
_ _
Given: CB ‖ ED, AB ‖ CD, and CB ≅ ED.
A
Prove: △ABC ≅ △CDE
CB ≅ ED
C
B
E
D
Given
CB ǁ ED
∠ACB ≅ ∠CED
△ABC ≅ △CDE
Given
Corr. ! Thm.
AAS Cong. Thm.
AB ǁ CD
∠CAB ≅ ∠ECD
Given
Corr. ! Thm.
Module 6
285
Lesson 2
COLLABORATIVE LEARNING
GE_MNLESE385795_U2M06L2 285
Small Group Activity
Students may benefit from exploring conditions that do not guarantee
congruent triangles. Have students work in small groups and use a protractor
to draw triangles with angles that measure 35°, 50°, and 95°. Then then compare
their triangles within each group. Students will see that all the triangles have the
same shape, but not necessarily the same size. Since the triangles are not all
congruent, the activity shows that there is no AAA Congruence Criteria.
285
Lesson 6.2
5/22/14 9:04 PM
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Your Turn
5.
EXPLAIN 3
_ _ _ _
Given: ∠ABC ≅ ∠DEF, BC ∥ EF, AC ≅ DF. Use the AAS Theorem to prove the
triangles are congruent.
Applying Angle-Angle-Side
Congruence
Write a paragraph proof.
E
B
AVOID COMMON ERRORS
C
A
F
D
The real-world situation may distract some students.
Make sure they are certain what they are solving for
before they begin.
¯ is parallel to EF
¯, this means that ∠ACB is congruent to ∠DFE,
Because BC
using the Corresponding Angles Theorem. Since ∠ABC is congruent to
¯ is congruent to DF
¯, then one pair of corresponding angles
∠DEF and AC
and two pairs of non-included corresponding sides are congruent.
This means that △ABC is congruent to △DEF using the AAS Triangle
Congruence Theorem.
Applying Angle-Angle-Side Congruence
Explain 3
The triangular regions represent plots of land. Use
the AAS Theorem to explain why the same amount of
fencing will surround either plot.
Example 3
Given: ∠A ≅ ∠D
It is given that ∠A ≅ ∠D. Also, ∠B ≅ ∠E
because both are right angles. Compare
AC and DF using the Distance Formula.
1
2
2
1
y
D
E
2
-2
0
2
F
4
x
-2
-4
2
© Houghton Mifflin Harcourt Publishing Company
2
A
-4
2
2
4
2
――――――――
AC = √(x - x ) + (y - y )
―――――――――
= √(-1-(-4)) + (4 - 0)
―――
= √3 + 4
―
= √25
2
C
B
=5
――――――――
―――――――
= √(4 - 0) + (1 - 4)
――――
= 4 + (-3)
―
= √25
DF =
√(x 2 - x 1) 2 + (y 2 - y 1) 2
2
2
2
2
=5
Because two pairs of angles and a pair of non-included sides are congruent,
△ABC ≅ △DEF by AAS. Therefore the triangles have the same perimeter and
the same amount of fencing is needed.
Module 6
286
Lesson 2
DIFFERENTIATE INSTRUCTION
GE_MNLESE385795_U2M06L2 286
Auditory Cues
6/16/15 10:23 AM
As students work through the lesson, have them identify orally whether the
triangles are congruent by ASA, AAS, SAS, or SSS. Then have them work with a
partner to identify the congruent parts and to determine if the congruent pairs of
angles or sides are corresponding parts.
AAS Triangle Congruence 286
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4.
QUESTIONING STRATEGIES
△GKJ and △JHG
G
H
5.
Yes. You cannot
use the HL Triangle K
J
Congruence
Theorem since it is not known
whether the triangles are right
triangles, but you can use the SSS
Triangle Congruence Theorem.
What type of triangle must be given to use
HL as a method of proof? It must be a right
triangle.
△EFG and △SQR
F
R
E
S
G
Q
Yes. You can use
Pythagorean Theorem
_
_ the
to show that FG ≅ QR and then use the HL
Triangle Congruence Theorem, or you can
use the SAS Triangle Congruence Theorem
with the given information.
Write a two-column proof, using the HL Congruence Theorem, to prove that the
triangles are congruent.
6.
―
―
Given: ∠A and ∠B are right angles. AB ≅ DC
Prove: △ABC ≅ △DCB
A
D
B
C
Statements
Reasons
1. ∠A and ∠B are right angles.
_ _
2. AB ≅ DC
_ _
3. BC ≅ BC
4. △ABC ≅ △DCB
7.
1. Given
2. Given
3. Reflexive Property of Congruence
4. HL Triangle Congruence Theorem
Given: ∠FGH and ∠JHK
are right
_ _
_ angles.
H is the midpoint of GK. FH ≅ JK
Prove: △FGH ≅ △JHK
G
© Houghton Mifflin Harcourt Publishing Company
Statements
2. H is the midpoint of GK.
_ _
3. GH ≅ HK
_ _
4. FH ≅ JK
3. Definition of midpoint
4. Given
5. △FGH ≅ △JHK
5. HL Triangle Congruence Theorem
―
2. Given
―
to QR.
Given: MP is perpendicular
―_ ―
N is the midpoint of MP. QP ≅ RM
Prove: △MNR ≅ △PNQ
M
_ _
MP ⊥ QR so ∠QNP and ∠MNR are right angles
(definition
of perpendicular). N is the midpoint
_
_
PN (definition
midpoint).
of MP, so MN ≅ ¯
_ of _
Then, since it is given that QP ≅ RM , △MNR ≅ △PNQ
by the HL Triangle Congruence Theorem.
Exercise
GE_MNLESE385795_U2M06L3.indd 299
Lesson 6.3
Reasons
1. Given
Module 6
299
K
H
1. ∠FGH and ∠JHK are right angles.
―
8.
J
F
Q
N
P
R
299
Lesson 3
Depth of Knowledge (D.O.K.)
Mathematical Practices
23/05/14 7:18 PM
22
3 Strategic Thinking
MP.2 Reasoning
23
3 Strategic Thinking
MP.2 Reasoning
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20°
ladd
er
The house is perpendicular to the ground,
so the other remote interior angle is 90°.
20 + 90 = 110, so the measure of the
indicated exterior angle is 110°.
house
18. A ladder propped up against a house makes a 20° angle with the
wall. What would be the ladder's angle measure with the ground
facing away from the house?
?
ground
19. Photography The aperture of a camera is made by overlapping blades that form a regular decagon.
a. What is the sum of the measures of the interior angles of
the decagon?
(10 - 2)180° = (8)180° = 1440°
b. What would be the measure of each interior angle? each
exterior angle?
1440° ÷ 10 = 144°; 180° - 144° = 36°
c.
Find the sum of all ten exterior angles.
36°(10) = 360°
20. Determine the measure of ∠UXW in the diagram.
m∠WUX = 90°
Y
V
© Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits:
©neyro2008/iStockPhoto.com
U
m∠UXW = 36°
W
Z
21. Determine the measures of angles x, y, and z.
x = 180 - (100 + 60) = 20°
80°
y = 180 - (80 + 55) = 45°
100°
55°
GE_MNLESE385795_U2M07L1.indd 323
Lesson 7.1
78°
54°
X
Module 7
323
180° = 54° + 90° + m∠UXW
x°
z°
y°
60°
z = 180 - (20 + 45) = 115°
323
Lesson 1
20/03/14 12:28 PM
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_
19. Critical Thinking Prove ∠B ≅ ∠C, given point M is the midpoint of BC.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Remind students that an isosceles triangle has
A
B
Statements
_
1. M is the midpoint of BC.
― ―
C
M
at least two congruent sides, and its properties can be
used to prove that it also has at least two congruent
angles.
Reasons
1. Given
2. BM ≅ CM
2. Definition of midpoint
― ―
4. AM ≅ AM
4. Reflexive Property of Congruence
5. △AMB ≅ △AMC
5. SSS Triangle Congruence Theorem
6. ∠B ≅ ∠C
6. CPCTC
― ―
3. AB ≅ AC
3. Given
_
_
20. Given that △ABC is an isosceles triangle and AD and CD are angle bisectors,
what is m∠ADC?
B
m∠BAC = m∠BCA = 70°, so m∠DAC = m∠DCA = 35°. Then,
m∠ADC = 180° − (35° + 35°) = 110°.
40°
D
A
C
H.O.T. Focus on Higher Order Thinking
The triangles are congruent isosceles triangles; the bisected right angle
results in two 45° angles, and the perpendicular segments result in two
right angles, so angles A and C must also measure 45°. Since ¯
BD ≅ ¯
BD by
the Reflexive Property, triangles ABD and CBD are congruent by ASA.
A
D
B
Module 7
GE_MNLESE385795_U2M07L2 338
© Houghton Mifflin Harcourt Publishing Company
21. Analyze
Isosceles right triangle_
ABC has
_ _Relationships
_ angle at B and
_
_ a right
AB ≅ CB. BD bisects angle B, and point D is on AC. If BD ⟘ AC, describe triangles ABD and CBD.
Explain. HINT: Draw a diagram.
C
338
Lesson 2
5/22/14 4:55 PM
Isosceles and Equilateral Triangles
338