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Name———————————————————————— Date —————————————
Challenge Practice
Lesson
4.7
For use with the lesson “Use Congruent Triangles”
In Exercises 1–6, refer to the diagram to write a two-column proof.
}
1. GIVEN: V is the midpoint of ​XZ​ .
2. GIVEN: ∠ 2 > ∠ 1, ∠ 4 > ∠ 5
}i}
XY​
​  
​   ZW​
​} ​ }
​  ​
PROVE: KL > ML
} }
​  
PROVE: ​YV​ > WV​
W
X
K
V
1
2
3
J
Z
4
5
L
Y
M
}
4. GIVEN: ∠ R > ∠ S, ∠ 2 > ∠ 3
} } } }
PJ​
​  , PL​
​  > QL​
​  , ∠ PKJ ​ ​  ​} ​
​  > QN​
}
3. GIVEN: L is the midpoint of JN​
​  ,
PROVE: RU > SU
and ∠ QMN are right angles.
PROVE: ∠ MQN > ∠ KPJ
P
R
S
1
U
5
2
6
3
4
T
J
K
}
L
} }
M
V
N
}
5. GIVEN: ​BC​ > CD​
​  , AB​
​  > AD​
​  
}
}
6. GIVEN: ​AB​ and CD​
​  bisect each other at
​}​  ​} ​
point M.
PROVE: AC ⊥ BD
} }
​  
PROVE: ​AD​ i BC​
B
A
E
D
C
Lesson 4.7
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
D
A
1
M
B
2
C
Geometry
Chapter Resource Book
4-99
Statements
Reasons
5. h 5 h1 5. Corresponding
s are >.
parts of > n
Study Guide
1. If you can show that n JKM > n LKM, then
} }
} }
​  . Since KM​
​  > KM​
​  by
you will know that JK​
​  > LK​
the reflexive property, then n JKM > n LKM
by the SAS Congruence Postulate. Because
corresponding parts of congruent triangles are
} }
​  .
congruent, JK​
​  > LK​
2. If you can show that n PQR > n RST, then
you will know that ∠ RPQ > ∠ TRS. Because
RQ i TS, ∠ PRQ > ∠ RTS by the Corresponding
Angles Postulate. By the AAS Congruence
Theorem, n PQR > n RST. Because
corresponding parts of congruent triangles are
congruent, ∠ RPQ > ∠ TRS. 3. Use the
SAS Congruence Postulate to prove that
n ABF > n EBC. Then state that
∠ AFB > ∠ ECB because they are
corresponding parts of congruent triangles.
∠ CBD and ∠ FBG are congruent because they
are vertical angles. Use the ASA Congruence
Postulate to prove that n FBG > n CBD. Then ​
} }
BG​ > BD​
​  because they are corresponding parts of
congruent triangles. 4. Use the SAS Congruence
Postulate to prove that n PTS > n PRS. Then state
} }
​  because they are corresponding
that PT​
​  > PR​
parts of congruent triangles. Use the SAS
Congruence Postulate to prove that
n PTU > n PRQ.
Real-Life Application
1.
Statements
1. ∠ ABD > ∠ CBD
2.∠BDA and ∠BDC are right angles
3. ∠ BDA > ∠ BDC } }
​  
4. ​DB​ > DB​
5. n ABD > nCBD 2.
sight mirror
angle
6 in.
eye
level
A60
reflection
angle
Reasons
1. Given
2. Given
3. Def. of a right angle
4. Reflexive Property
5. ASA Congruence
Postulate
3. Lower it 3 inches.
sight mirror
angle
3 in.
3 in.
eye
level
3 in.
reflection
angle
Challenge Practice
1.
Statements
Reasons
1. ∠ WVZ > ∠ YVX 1. Vertical Angles
Theorem
2. V is the midpoint 2. Given
}
of ​XZ​ .
}
}
​  
​  > XV​
3. Definition of midpoint
3. ZV​
}i}
​  
4. Given
4. ​XY​  ZW​
5. ∠ ZWV > ∠ XYV 5. Alt. Int. Angles
Theorem
6. n WVZ > n YVX 6. AAS Congruence
Theorem
} }
​  
7. Corresp. parts of > n
7. YV​
​  > WV​
are >.
2.
Statements
Reasons
1. ∠ 2 > ∠ 1, 1. Given
∠ 4 > ∠ 5
2. ∠ 1 > ∠ 3 2. Vertical Angles Thm
3. m∠ 2 5 m∠ 1, 3. Definition of
m∠ 1 5 m∠ 3 congruent angles
4. m∠ 2 5 m∠ 3 4. Substitution property
of equality
5. ∠ 2 > ∠ 3 5. Definition of
congruent angles
} }
​  
6. Reflexive property
6. JL​
​  > JL​
of congruence
7. n JKL > n JML
7. ASA Cong Post
} }
​  
8. Corresp. parts of > n
8. ​KL​ > ML​
are >.
3.
Statements
1. L is the midpoint }
of JN​
​  .
} }
​  
2. JL​
​  > LN​
} }
​  ,
3. PJ​
​  > QN​
} }
​  
​PL​ > QL​
4. n JLP > n NLQ
5. ∠ PKJ and ∠ QMN are right angles.
Reasons
1. Given
2. Definition of midpoint
3. Given
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
Lesson 4.7 Use Congruent
Triangles, continued
4. SSS Cong Post
5. Given
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C4AK.indd 60
4/28/11 6:14:13 PM
Lesson 4.7 Use Congruent
Triangles, continued
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
4.
Statements
Reasons
1. ∠ R > ∠ S, 1. Given
∠ 2 > ∠ 3
} }
​  
2. Reflexive property of
2. TV​
​  > TV​
congruence
3. n RTV > n SVT 3. AAS Congruence
Theorem
} }
4. RT​
​  > ​SV​  4. Corresp. parts of > n
are >.
5. ∠ 5 > ∠ 6 5. Vertical Angles
Theorem
6. n RTU > n SVU 6. AAS Congruence
Theorem
} }
7. RU​
​  > ​SU​  7. Corresp. parts of > n
are >.
5.
Statements Reasons
} }
​  , 1. Given
1. BC​
​  > CD​
} }
AB​
​  
​  > AD​
} }
​   2. Reflexive property
2. CA​
​  > CA​
of congruence
3. n ABC > n ADC 3. SSS Congruence
Postulate
4. ∠ BCA > ∠ DCA 4. Corresp. parts of >
n are >.
} }
​   5. Reflexive property
5. CE​
​  > CE​
of congruence
6. n CEB > n CED 6. SAS Congruence
Postulate
7. ∠ CEB > ∠ CED 7. Corresp. parts of >
n are >.
8. m∠ CEB 5 8. Definition of
m∠ CED congruent angles
9. m∠ CEB 1 9. Linear Pair Postulate
m∠ CED 5 1808 10. m∠ CEB 1 10. Substitution
m∠ CEB 5 1808 property of equality
6.
Statements
Reasons
}
}
​  bisect 1. Given
1. AB​
​  and CD​
each other at point M.
2. M
is the midpoint 2. Definition of
}
}
​  and CD​
​  . segment bisector
of AB​
} }
​  , 3. Definition of
3. AM​
​  > MB​
} }
​   midpoint
​DM​ > MC​
4. ∠ AMD > ∠ BMC 4. Vertical Angles
Theorem
5. n AMD > n BMC 5. SAS Congruence
Postulate
6. ∠ A > ∠ B 6. Corresp. parts
of > n are >.
} }
​   7. Alternate Interior
7. ​AD​ i BC​
Angles Theorem
Lesson 4.8 Use Isosceles and
­Equilateral Triangles
Teaching Guide
1. Check drawings. 2. (a) no congruent angles
(b) two congruent angles (c) three congruent
angles
3. If a triangle has n congruent sides, then it has
n congruent angles. 4. If a triangle has n
congruent angles, then it has n congruent sides.
Yes. A triangle has n congruent sides if and only if
it has n congruent angles.
Investigating Geometry Activity
1. They are congruent. 2. If two sides of a
triangle are congruent, then the angles opposite
them are congruent. 3. Yes, nDEF is isosceles
}
}
​  are congruent. 4. If two
because ​DE​ and EF​
angles of a triangle are congruent, then the sides
opposite them are congruent.
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C4AK.indd 61
answers
Statements Reasons
6. ∠ PKJ > ∠ QMN 6. Right Angles
Congruence Theorem
7. ∠ KJP > ∠ MNQ 7. Corresp. parts of > n
are >.
8. n PKJ > n QMN 8. AAS Cong Thm
9. ∠ MQN > ∠ KPJ 9. Corresp. parts of > n
are >.
Statements Reasons
11. 2m∠CEB 5 1808 11. Simplify.
12. m∠ CEB 5 908 12. Division property of
equality
13. ∠ CEB and ∠ CED 13. Definition of
are right angles. right angle
} }
14. Definition of
14. ​AC​ ⊥ ​BD​ 
perpendicular lines
A61
4/28/11 6:14:13 PM
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