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Using Congruent Triangles (PowerPoint)

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					4.5 Using Congruent
     Triangles
Objectives/Assignment
 Use congruent triangles to plan and write
 proofs
 Use congruent triangles to prove
 constructions are valid
 Assignment: 5-19 odd, 20, 21
Goal 1: Planning a Proof
 Knowing that all pairs of corresponding
 parts of congruent triangles are congruent
 can help you reach conclusions about
 congruent figures.
Planning a Proof
 For example, suppose you
 want to prove that PQS ≅            Q
 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           R
 corresponding       parts   of   P
 congruent     triangles    are
 congruent to conclude that           S
 PQS ≅ RQS.
Example 1: Planning & Writing a
Proof
 Given: AB ║ CD, BC ║
 AD                            B       C
 Prove: AB≅CD
 Plan for proof: Show
 that ∆ABD ≅ ∆CDB.
 Then use the fact that
 corresponding parts of    A       D
 congruent triangles are
 congruent.
Example 1: Planning & Writing a
Proof
 Solution:      First copy the
 diagram and mark it with the            B       C
 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     A       D
 intersected by a transversal, you
 can deduce that two pairs of
 alternate interior angles are
 congruent.
Example 1: Paragraph Proof
Because AD ║CD, it follows
from the Alternate Interior
Angles Theorem that ABD
                                      B       C
≅CDB.       For the same
reason,    ADB        ≅CBD
because BC║DA.          By the
Reflexive      property      of
Congruence, BD ≅ BD. You
can use the ASA Congruence
Postulate to conclude that        A
∆ABD ≅ ∆CDB.            Finally           D
because corresponding parts
of congruent triangles are
congruent, it follows that AB
≅ CD.
Example 2: Planning & Writing a
Proof
 Given: A is the midpoint of
 MT, A is the midpoint of SR.
                                M       R
 Prove: MS ║TR.
 Plan for proof: Prove that
 ∆MAS ≅ ∆TAR. Then use
 the fact that corresponding        A
 parts of congruent triangles
 are congruent to show that
 M ≅ T. Because these                 T
 angles are formed by two       S
 segments intersected by a
 transversal, you can
 conclude that MS ║ TR.
                                                   M                 R
Given: A is the midpoint of MT, A is the
midpoint of SR.
Prove: MS ║TR.                                             A

                                                   S                 T
Statements:                           Reasons:
1.   A is the midpoint of MT, A       1.   Given
     is the midpoint of SR.
2.   MA ≅ TA, SA ≅ RA                 2.   Definition of a midpoint
3.   MAS ≅ TAR                      3.   Vertical Angles Theorem
4.   ∆MAS ≅ ∆TAR                      4.   SAS Congruence Postulate
5.   M ≅ T                          5.   Corres. parts of ≅ ∆’s are ≅
6.   MS ║ TR                          6.   Alternate Interior Angles
                                           Converse
Example 3: Using more than one
pair of triangles
 Given: 1≅2, 3≅4
 Prove ∆BCE≅∆DCE                               D
 Plan for proof:         The only
 information you have about
 ∆BCE and ∆DCE is that 1≅2              2                4
 and that CE ≅CE.           Notice,   C                3       A
 however, that sides BC and DC             1       E
 are also sides of ∆ABC and
 ∆ADC. If you can prove that
 ∆ABC≅∆ADC, you can use the                    B
 fact that corresponding parts of
 congruent       triangles     are
 congruent to get a third piece of
 information about ∆BCE and
 ∆DCE.
Given: 1≅2, 3≅4.             2                  4
Prove ∆BCE≅∆DCE        C        1           E       3        A

                                        B
Statements:                Reasons:
1. 1≅2, 3≅4            1. Given
2. AC ≅ AC                 2. Reflexive property of
                              Congruence
3. ∆ABC ≅ ∆ADC             3. ASA Congruence
                              Postulate
                           4.        Corres. parts of ≅ ∆’s are ≅
4. BC ≅ DC
                           5.        Reflexive Property of
5. CE ≅ CE                           Congruence
6. ∆BCE≅∆DCE               6.        SAS Congruence Postulate
Goal 2: 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.
Plan for Proof
 Show that ∆CAB ≅ ∆FDE.                C
 Then use the fact that
 corresponding      parts     of
 congruent     triangles    are
 congruent to conclude that        A
 CAB ≅ FDE.                By
 construction,     you      can        B
 assume       the     following        F
 statements:
  – AB ≅ DE Same compass
    setting is used
  – AC ≅ DF Same compass           D
    setting is used
  – BC ≅ EF Same compass               E
    setting is used
                                         C


Given: AB ≅ DE, AC ≅ DF, BC ≅ EF     A

Prove CAB≅FDE                          B
                                         F



                                     D

                                         E



Statements:                    Reasons:
1.   AB ≅ DE                   1.   Given
2.   AC ≅ DF                   2.   Given
3.   BC ≅ EF                   3.   Given
4.   ∆CAB ≅ ∆FDE               4.   SSS Congruence Post
5.   CAB ≅ FDE               5.   Corres. parts of ≅ ∆’s
                                    are ≅.

				
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posted:9/20/2011
language:English
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