# 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
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 ║
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.
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
from the Alternate Interior
Angles Theorem that ABD
B       C
≅CDB.       For the same
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
∆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
∆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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 views: 44 posted: 9/20/2011 language: English pages: 14