4.5 Using Congruent Triangles

1
4.5 Using Congruent Triangles
2
Objectives

• Use congruent triangles to plan and write proofs.
• Use congruent triangles to prove constructions
  are valid.

3
Planning a proof

• Knowing that all pairs of corresponding parts of
  congruent triangles are congruent can help you
  reach conclusions about congruent figures.

4
Planning a proof

• For example, suppose you want to prove that ?PQS
  ? ?RQS in the diagram shown at the right. One
  way to do this is to show that ?PQS ? ?RQS by the
  SSS Congruence Postulate. Then you can use the
  fact that corresponding parts of congruent
  triangles are congruent to conclude that ?PQS ?
  ?RQS.

5
Ex. 1 Planning Writing a Proof

• Given AB CD, BC DA
• Prove AB?CD
• Plan for proof Show that ?ABD ? ?CDB. Then use
  the fact that corresponding parts of congruent
  triangles are congruent.

6
Ex. 1 Planning Writing a Proof

• Solution First copy the diagram and mark it
  with the given information. Then mark any
  additional information you can deduce. Because
  AB and CD are parallel segments intersected by a
  transversal, and BC and DA are parallel segments
  intersected by a transversal, you can deduce that
  two pairs of alternate interior angles are
  congruent.

7
Ex. 1 Paragraph Proof

• Because AD CD, it follows from the Alternate
  Interior Angles Theorem that ?ABD ??CDB. For the
  same reason, ?ADB ??CBD because BCDA. By the
  Reflexive property of Congruence, BD ? BD. You
  can use the ASA Congruence Postulate to conclude
  that ?ABD ? ?CDB. Finally because corresponding
  parts of congruent triangles are congruent, it
  follows that AB ? CD.

8
Ex. 2 Planning Writing a Proof

• Given A is the midpoint of MT, A is the
  midpoint of SR.
• Prove MS TR.
• Plan for proof Prove that ?MAS ? ?TAR. Then
  use the fact that corresponding parts of
  congruent triangles are congruent to show that ?M
  ? ?T. Because these angles are formed by two
  segments intersected by a transversal, you can
  conclude that MS TR.

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Given A is the midpoint of MT, A is
themidpoint of SR.Prove MS TR.

• Statements
• A is the midpoint of MT, A is the midpoint of SR.
• MA ? TA, SA ? RA
• ?MAS ? ?TAR
• ?MAS ? ?TAR
• ?M ? ?T
• MS TR

• Reasons
• Given

10
Given A is the midpoint of MT, A is
themidpoint of SR.Prove MS TR.

• Statements
• A is the midpoint of MT, A is the midpoint of SR.
• MA ? TA, SA ? RA
• ?MAS ? ?TAR
• ?MAS ? ?TAR
• ?M ? ?T
• MS TR

• Reasons
• Given
• Definition of a midpoint

11
Given A is the midpoint of MT, A is
themidpoint of SR.Prove MS TR.

• Statements
• A is the midpoint of MT, A is the midpoint of SR.
• MA ? TA, SA ? RA
• ?MAS ? ?TAR
• ?MAS ? ?TAR
• ?M ? ?T
• MS TR

• Reasons
• Given
• Definition of a midpoint
• Vertical Angles Theorem

12
Given A is the midpoint of MT, A is
themidpoint of SR.Prove MS TR.

• Statements
• A is the midpoint of MT, A is the midpoint of SR.
• MA ? TA, SA ? RA
• ?MAS ? ?TAR
• ?MAS ? ?TAR
• ?M ? ?T
• MS TR

• Reasons
• Given
• Definition of a midpoint
• Vertical Angles Theorem
• SAS Congruence Postulate

13
Given A is the midpoint of MT, A is
themidpoint of SR.Prove MS TR.

• Statements
• A is the midpoint of MT, A is the midpoint of SR.
• MA ? TA, SA ? RA
• ?MAS ? ?TAR
• ?MAS ? ?TAR
• ?M ? ?T
• MS TR

• Reasons
• Given
• Definition of a midpoint
• Vertical Angles Theorem
• SAS Congruence Postulate
• Corres. parts of ? ?s are ?

14
Given A is the midpoint of MT, A is
themidpoint of SR.Prove MS TR.

• Statements
• A is the midpoint of MT, A is the midpoint of SR.
• MA ? TA, SA ? RA
• ?MAS ? ?TAR
• ?MAS ? ?TAR
• ?M ? ?T
• MS TR

• Reasons
• Given
• Definition of a midpoint
• Vertical Angles Theorem
• SAS Congruence Postulate
• Corres. parts of ? ?s are ?
• Alternate Interior Angles Converse.

15
Ex. 3 Using more than one pair of triangles.

• Given ?1??2, ?3??4.
• Prove ?BCE??DCE
• Plan for proof The only information you have
  about ?BCE and ?DCE is that ?1??2 and that CE
  ?CE. Notice, however, that sides BC and DC are
  also sides of ?ABC and ?ADC. If you can prove
  that ?ABC??ADC, you can use the fact that
  corresponding parts of congruent triangles are
  congruent to get a third piece of information
  about ?BCE and ?DCE.

2
4
3
1
16
Given ?1??2, ?3??4.Prove ?BCE??DCE
4
2
3
1

• Statements
• ?1??2, ?3??4
• AC ? AC
• ?ABC ? ?ADC
• BC ? DC
• CE ? CE
• ?BCE??DCE

• Reasons
• Given

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Given ?1??2, ?3??4.Prove ?BCE??DCE
4
2
3
1

• Statements
• ?1??2, ?3??4
• AC ? AC
• ?ABC ? ?ADC
• BC ? DC
• CE ? CE
• ?BCE??DCE

• Reasons
• Given
• Reflexive property of Congruence

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Given ?1??2, ?3??4.Prove ?BCE??DCE
4
2
3
1

• Statements
• ?1??2, ?3??4
• AC ? AC
• ?ABC ? ?ADC
• BC ? DC
• CE ? CE
• ?BCE??DCE

• Reasons
• Given
• Reflexive property of Congruence
• ASA Congruence Postulate

19
Given ?1??2, ?3??4.Prove ?BCE??DCE
4
2
3
1

• Statements
• ?1??2, ?3??4
• AC ? AC
• ?ABC ? ?ADC
• BC ? DC
• CE ? CE
• ?BCE??DCE

• Reasons
• Given
• Reflexive property of Congruence
• ASA Congruence Postulate
• Corres. parts of ? ?s are ?

20
Given ?1??2, ?3??4.Prove ?BCE??DCE
4
2
3
1

• Statements
• ?1??2, ?3??4
• AC ? AC
• ?ABC ? ?ADC
• BC ? DC
• CE ? CE
• ?BCE??DCE

• Reasons
• Given
• Reflexive property of Congruence
• ASA Congruence Postulate
• Corres. parts of ? ?s are ?
• Reflexive Property of Congruence

21
Given ?1??2, ?3??4.Prove ?BCE??DCE
4
2
3
1

• Statements
• ?1??2, ?3??4
• AC ? AC
• ?ABC ? ?ADC
• BC ? DC
• CE ? CE
• ?BCE??DCE

• Reasons
• Given
• Reflexive property of Congruence
• ASA Congruence Postulate
• Corres. parts of ? ?s are ?
• Reflexive Property of Congruence
• SAS Congruence Postulate

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Ex. 4 Proving constructions are valid

• In Lesson 3.5 you learned to copy an angle
  using a compass and a straight edge. The
  construction is summarized on pg. 159 and on pg.
  231.
• Using the construction summarized above, you can
  copy ?CAB to form ?FDE. Write a proof to verify
  the construction is valid.

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Plan for proof

• Show that ?CAB ? ?FDE. Then use the fact that
  corresponding parts of congruent triangles are
  congruent to conclude that ?CAB ? ?FDE. By
  construction, you can assume the following
  statements
• AB ? DE Same compass setting is used
• AC ? DF Same compass setting is used
• BC ? EF Same compass setting is used

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Given AB ? DE, AC ? DF, BC ? EF Prove
?CAB??FDE
4
2
3
1

• Statements
• AB ? DE
• AC ? DF
• BC ? EF
• ?CAB ? ?FDE
• ?CAB ? ?FDE

• Reasons
• Given

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Given AB ? DE, AC ? DF, BC ? EF Prove
?CAB??FDE
4
2
3
1

• Statements
• AB ? DE
• AC ? DF
• BC ? EF
• ?CAB ? ?FDE
• ?CAB ? ?FDE

• Reasons
• Given
• Given

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Given AB ? DE, AC ? DF, BC ? EF Prove
?CAB??FDE
4
2
3
1

• Statements
• AB ? DE
• AC ? DF
• BC ? EF
• ?CAB ? ?FDE
• ?CAB ? ?FDE

• Reasons
• Given
• Given
• Given

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Given AB ? DE, AC ? DF, BC ? EF Prove
?CAB??FDE
4
2
3
1

• Statements
• AB ? DE
• AC ? DF
• BC ? EF
• ?CAB ? ?FDE
• ?CAB ? ?FDE

• Reasons
• Given
• Given
• Given
• SSS Congruence Post

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Given AB ? DE, AC ? DF, BC ? EF Prove
?CAB??FDE
4
2
3
1

• Statements
• AB ? DE
• AC ? DF
• BC ? EF
• ?CAB ? ?FDE
• ?CAB ? ?FDE

• Reasons
• Given
• Given
• Given
• SSS Congruence Post
• Corres. parts of ? ?s are ?.

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Given QS?RP, PT?RT Prove PS? RS
4
2
3
1

• Statements
• QS ? RP
• PT ? RT

• Reasons
• Given
• Given