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Name ——————————————————————— Date ————————————
List all of the pairs of angles and sides that are congruent based on the given congruence statement and the figure.
1. n RST > n XZY 2. n ABC > n DEF
R X
S Z
T Y
A
C B
E
D
F
Tell which triangles you can show are congruent in order to prove the statement. What postulate or theorem would you use?
3. ∠ C > ∠ F 4. }
RT ù } LN 5. ∠ D ù ∠ H
A
B C
D
E F
R
S T
N
L M E G
C F
D
H
6. ∠ STV > ∠ UTV 7. }
EF ù } KJ 8. }
XY ù } ZW
T
V
S U
E F
J K
D
X Y
W Z
Is enough information given in the figure to show that the given statement is true? Explain.
9. ∠ N > ∠ Q 10. } RU > } TU 11. }
FG > } HE
M N
P
R
R S
U
T
G
H
F
E
Practice AFor use with the lesson “Use Congruent Triangles”
Less
on
4.7
GeometryChapter Resource Book 4-89
Lesson
4.7
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Name ——————————————————————— Date ————————————
Use the diagram to write a plan for a proof.
12. PROVE: ∠ J ù ∠ N 13. PROVE: } ST ù } UQ
L J
P N
K
R
S
T
U
R
Use the information given in the diagram to write a proof.
14. PROVE: ∠ ABD ù ∠ CBD 15. PROVE: } UV ù } WX
B
D A C
V
U W
X
16. Using angles You can position yourself halfway between two buildings of equal height by moving to a position where congruent angles are formed between the horizontal and your line of sight to the top of each building. Verify this by completing the three step proof below.
GIVEN: } AB ù } ED , ∠ ACB > ∠ ECD,
A
B
E
D
C
∠ A and ∠ E are right angles.
PROVE: } AC ù } EC
Statements Reasons
1. } AB ù } ED , ∠ ACB > ∠ ECD, 1. ? ∠ A and ∠ E are right angles.
2. n ABC > n EDC 2. ?
3. } AC ù } EC 3. ?
Practice A continuedFor use with the lesson “Use Congruent Triangles”
Les
so
n 4
.7
GeometryChapter Resource Book4-90
Lesson
4.7
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Statements Reasons
6. AC 5 CD 1 BC 6. Substitution property of equality
7. AC 5 BD 7. Substitution property of equality
8. } AC > }
BD 8. Definition of congruent segments
9. ∠ FDB > ∠ ECA 9. Corresponding Angles Postulate
10. n AEC > n BFD 10. ASA Congruence Postulate
6.
Statements Reasons
1. ∠ KNL > ∠ MNL, 1. Given ∠ KLN > ∠ MLN
2. } NL > }
NL 2. Reflexive property of congruence
3. n KNL > n MNL 3. ASA Congruence Postulate
4. } NK > }
NM 4. Corresponding parts of congruent triangles are congruent
5. m∠ JNK 1 m∠ KNL 5. Linear Pair 5 1808, Postulate m∠ JNM 1 m∠ MNL 5 1808
6. m∠ JNK 1 m∠ KNL 6. Transitive 5 m∠ JNM 1 property of m∠ MNL equality
7. m∠ KNL 5 m∠ MNL 7. Definition of congruent angles
8. m∠ JNK 1 m∠ KNL 8. Substitution 5 m∠ JNM 1 property of m∠ KNL equality
9. m∠ JNK 5 m∠ JNM 9. Subtraction property of equality
10. ∠ JNK > ∠ JNM 10. Definition of congruent angles
11. } JN > }
JN 11. Reflexive property of congruence
12. n JNK > n JNM 12. SAS Congruence Postulate
Lesson Use Congruent TrianglesTeaching Guide
1. SAS (all congruence assumed); corresponding parts of congruent triangles are congruent.
2.
SAS (one pair of sides congruent by reflexive property, one pair of sides assumed congruent, angles assumed congruent); corresponding parts of congruent triangles are congruent.
3. SAS (all congruence assumed); corresponding parts of congruent triangles are congruent.
4. SAS (both pairs sides assumed con-gruent, vertical angles congruent); corresponding parts of congruent triangles are congruent.
5.
SAS (one pair of sides congruent by reflexive property, one pair of sides assumed congruent, angles assumed congruent); corresponding parts of congruent triangles are congruent.
6. SAS (one pair of sides congruent by reflexive property, one pair of sides assumed congruent, angles assumed congruent); corresponding parts of congruent triangles are congruent.
Practice Level A
1. ∠ R > ∠ X, ∠ S > ∠ Z, ∠ T > ∠ Y, }
RS > } XZ , }
ST > } ZY , } RT > } XY 2. ∠ A > ∠ D, ∠ B > ∠ E, ∠ C > ∠ F,
} AB > } DE ,
} BC > } EF ,
}
AC > } DF 3. n ABC > n DEF; HL
4. n RST > n NML; AAS 5. n CDE > n FHG; ASA 6. n STV > n UTV; SSS 7. n DEF > n DKJ; SAS 8. XYZ > n ZWX; ASA
9. Yes; n MNP > n RQP by ASA, so ∠ N and ∠ Q are corresponding parts of > ns .
10. No; Only 2 pairs of sides can be assumed to be > in n RSU and n TSU, so there is not enough information to use congruent triangles.
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Lesson Prove Triangles Congruent by ASA and AAS, continued
GeometryChapter Resource Book A57
4.6 4.7
CS10_CC_G_MECR710761_C4AK.indd 57 4/28/11 6:14:11 PM
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11. Yes; n EFH > n GHF by AAS, so }
FG and } HE are corresponding parts of > ns . 12. Use SSS to prove n JKL > n NRP, then use the fact that ∠ J and ∠ N are corresponding parts of > ns .
13. Show because vertical angles, ∠ SRT > ∠ URQ. Use AAS to show n RST > n RUQ, then use the fact that
} ST
and }
UQ are corresponding parts of > ns
14. Statements Reasons
1. } AD > }
CD , } BD ⊥ }
AC 1. Given2. ∠ ADB and ∠ CDB 2. Thm 3.9
are right angles.3. ∠ ADB > ∠ CDB 3. All right angles
are >.4. } BD > } BD 4. Reflexive Prop. of
Congruence 5. n ADB > n CBD 5. SAS Congruence
Post.6. ∠ ABD > ∠ CBD 6. Corr. parts of > ns
are >. 15. Statements Reasons
1. } VW i } XU , ∠ VUW 1. Given and ∠ XWU are right angles.
2. } UW > } WU 2. Reflexive Prop. of Congruence
3. ∠ VUW > ∠ XWU 3. All right ? are >.4. ∠ VWU > ∠ XUW 4. Alt. Interior Angles
Thm.5. n UVW > n WXU 5. ASA Congruence
Post.6. } UV > } WX 6. Corr. parts of > ns
are >.16. Given; AAS Congruence Theorem; Corresponding parts of > ns are >.
Practice Level B
1. n ABC ù n CDA; SAS
2. n TSU ù n VSU; AAS
3. n ABD ù nC DB; SSS
4. n NKH ù n TMG; AAS
5. n ABD ù n CBE; ASA
6. n ABC ù n STA; AAS
7. Use the HL Congruence Theorem to prove that n DAB ù n BCD. Then use the fact that
corresponding parts of congruent triangles are congruent to prove that ∠ DAB ù ∠ BCD.
8. Because }
ST i } RQ , ∠ PRQ ù / RST by the Corresponding Angles Postulate. Use the ASA Congruence Postulate to prove that n PRQ ù n RST. Then use the fact that corresponding parts of congruent triangles are congruent to prove that
} ST ù
} RQ .
9. Use the Distance Formula to find the side lengths of the triangles. Use the SSS Congruence Postulate to show that n ABC ù n DEF. Then use the fact that corresponding parts of congruent triangles are congruent to prove that ∠ A ù ∠ D.
10. Use the Distance Formula to find the side lengths of the triangles. Use the SSS Congruence Postulate to show that n ABC ù n DEF. Then use the fact that corresponding parts of congruent triangles are congruent to prove that ∠ A ù ∠ D.
11. Given; Given; Definition of angle bisector; Reflexive Property of Congruence; SAS Congruence Postulate; Corresponding parts of congruent triangles are congruent.
12.
Statements Reasons
1. } MQ ù } NT 1. Given2. } MQ i } NT 2. Given3. ∠ NTM ù ∠ QMT 3. Alternate Interior
Angles Theorem4. } MT ù } MT 4. Reflexive Property of
Congruence5. n NTM ù n QMT 5. SAS Congruence
Postulate6. } MN ù
} TQ 6. Corresponding parts
of congruent triangles are congruent.
13.
Statements Reasons
1. } AB ù } BE 1. Given2. ∠ ADB ù ∠ ECB 2. Given3. ∠ ABD ù ∠ EBC 3. Vertical Angles
Theorem4. n ABD ù n EBC 4. AAS Congruence
Theorem5. } DB ù
} CB 5. Corresponding parts
of congruent triangles are congruent.
Practice Level C
1. n HGL > n JKM; AAS 2. n PQU > n VPS; AAS 3. n ABC > n DEF; ASA
Lesson Use Congruent Triangles, continued
GeometryChapter Resource BookA58
4.7
CS10_CC_G_MECR710761_C4AK.indd 58 4/28/11 6:14:12 PM
