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Proving Triangles Congruent - Po

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Proving Triangles Congruent - Po Powered By Docstoc
					Proving Triangles
   Congruent
The Idea of a Congruence

Two geometric figures with
exactly the same size and
shape.
                      F


       B



       A     C    E   D
How much do you
   need to know. . .

   . . . about two triangles
         to prove that they
         are congruent?
        Corresponding Parts
In Lesson 4.2, you learned that if all
  six pairs of corresponding parts (sides
  and angles) are congruent, then the
  triangles are congruent.
    1. AB  DE
    2. BC  EF
    3. AC  DF
    4.  A   D
                            ABC   DEF
    5.  B   E
    6.  C   F
Do you need all six ?

          NO !

                   SSS
                   SAS
                   ASA
                   AAS
   Side-Side-Side (SSS)




1. AB  DE
2. BC  EF            ABC   DEF
3. AC  DF
     Postulate 19: SSS Congruence Postulate
EXAMPLE 3     Solve a real-world problem

 Structural Support

 Explain why the bench with the diagonal support is
 stable, while the one without the support can collapse.
EXAMPLE 3     Solve a real-world problem

 SOLUTION

 The bench with a diagonal support forms triangles with
 fixed side lengths. By the SSS Congruence Postulate,
 these triangles cannot change shape, so the bench is
 stable. The bench without a diagonal support is not
 stable because there are many possible quadrilaterals
 with the given side lengths.
EXAMPLE 2        Standardized Test Practice




  SOLUTION
 By counting, PQ = 4 and QR = 3. Use the Distance Formula
 to find PR.
         d   =     ( x2 – x1 ) 2 + ( y2 – y1 ) 2
EXAMPLE 2            Standardized Test Practice

            PR =       ( – 1 – (– 5 ) )2 + ( 1 – 4 ) 2

                 =      4 2 + (– 3 ) 2 =      25 = 5

 By the SSS Congruence Postulate, any triangle with
 side lengths 3, 4, and 5 will be congruent to PQR.
 The distance from (–1, 1) to (–1, 5) is 4. The distance from
 (–1, 5) to (–4, 5) is 3. The distance from (– 1, 1) to (–4, 5) is
   ( 5 – 1) 2 + ( (–4) – (–1) ) 2 =      4 2 + (– 3 ) 2 =   25 = 5


 ANSWER The correct answer is A.
EXAMPLE 1         Use the SSS Congruence Postulate

 Write a proof.

GIVEN    KL       NL, KM    NM

PROVE         KLM          NLM

 Proof    It is given that KL     NL and KM     NM
          By the Reflexive Property, LM       LM.
          So, by the SSS Congruence
          Postulate,   KLM     NLM
GUIDED PRACTICE       for Example 1

 Decide whether the congruence statement is true.
 Explain your reasoning.

 1.     DFG        HJK


 SOLUTION
 Three sides of one triangle are congruent to three
 sides of second triangle then the two triangle are
 congruent.
 Side DG   HK, Side DF   JH,and Side FG JK.
 So by the SSS Congruence postulate,     DFG          HJK.
 Yes. The statement is true.
GUIDED PRACTICE       for Example 1

 Decide whether the congruence statement is true.
 Explain your reasoning.

 2.     ACB         CAD


 SOLUTION

 GIVEN : BC    AD
 PROVE :      ACB     CAD
 PROOF:    It is given that BC AD By Reflexive property
           AC AC, But AB is not congruent CD.
  Side-Angle-Side (SAS)
       Postulate 20



1. AB  DE
2. A   D    ABC   DEF
3. AC  DF
              included
                angle
      Included Angle
The angle between two sides




 G          I           H
        Included Angle

           Name the included angle:
    E

           YE and ES     E
           ES and YS     S

Y   S      YS and YE     Y
EXAMPLE 2        Prove the AAS Congruence Theorem

 Prove the Angle-Angle-Side Congruence Theorem.

Write a proof.

GIVEN       A         D,   C     F, BC   EF

 PROVE          ABC        DEF
GUIDED PRACTICE         for Examples 1 and 2

 1.    In the diagram at the right, what
       postulate or theorem can you use to
       prove that     RST       VUT ? Explain.




 SOLUTION
      STATEMENTS                   REASONS
             S      U                Given
            RS    UV                 Given
          RTS       UTV              The vertical angles
                                     are congruent
              Do Now
           Eliminate the
           possibilities….
• Some of the measurements of ABC
  and DEF are given below. Can you
  determine if the two triangles are
  congruent from this information?
                                        2.5

                  2.5 cm
      30                     30

           4 cm                   4cm
  Angle-Side-Angle (ASA)
       Postulate 21



1. A   D
2. AB  DE     ABC   DEF
3.  B   E
               include
                   d
                 side
      Included Side
The side between two angles




 GI          HI               GH
        Included Side

          Name the included angle:
    E

          Y and E   YE
          E and S   ES
Y   S     S and Y   SY
  Angle-Angle-Side (AAS)
       Theorem 4.6



1. A   D
2.  B   E   ABC   DEF
3. BC  EF
               Non-included
                   side
    Angle-Angle-Side (AAS)
    • If two angles and a nonincluded side
       of a triangle are congruent to two
        angles and a nonincluded side of
         another triangle, then the two
            triangles are congruent.
       O                 A




            Y
B                 M           N
   Angle-Side-Side
• THERE IS NO SUCH THING
  AS ANGLE SIDE SIDE
  BECAUSE YOU CAN’T USE
  THAT KIND OF LANGUAGE
  AT SCHOOL.
Warning: No SSA Postulate

          There is no such
          thing as an SSA
             postulate!



      B               E

                             F
      A      C
                      D

          NOT CONGRUENT
Warning: No AAA Postulate
         There is no such
         thing as an AAA
            postulate!



                E
   B



   A     C                  F
                D

        NOT CONGRUENT
Tell whether you can use the
given information at
determine whether
ABC   DEF

A  D, ABDE, ACDF

AB  EF, BC  FD, AC DE
The Congruence Postulates
       SSS correspondence

       ASA correspondence

       SAS correspondence

       AAS   correspondence
       SSA correspondence

       AAA correspondence
Name That Postulate
               (when possible)




SAS
                   ASA




 SSA                 SSS
 Name That Postulate
                (when possible)




AAA
                       ASA



   SAS                     SSA
        Name That Postulate
                       (when possible)




                                   Vertical
                                   Angles
Reflexive
Property    SAS                    SAS




        Vertical           Reflexive
        Angles     SAS     Property    SSA
HW: Name That Postulate
                  (when possible)
HW: Name That Postulate
                  (when possible)
             Let’s Practice
 Indicate the additional information needed
 to enable us to apply the specified
 congruence postulate.

For ASA:   B  D
For SAS:   AC  FE
For AAS:   A  F
                 Closure
 Indicate the additional information needed
 to enable us to apply the specified
 congruence postulate.

For ASA:

For SAS:

For AAS:
Now For The Fun Part…
                          J
 Given: JO  SH;
O is the midpoint of SH
 Prove:  SOJ  HOJ
                  S       0   H
Write a two column Proof
Given: BC bisects AD and A   D
Prove: AB  DC
                      A             C
                            E
                      B             D

				
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