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4.4_asa_and_aas by dandanhuanghuang

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									                Geometry

            4.4 Proving Triangles Congruent
Worksheet
  4.4A


                                    mbhaub@mpsaz.org
Goals
 Use ASA and AAS postulates to prove two
  triangles are congruent.
 Solve problems using congruence
  postulates.




March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   2
Included Sides                               AC is included
         C                                   between A and C.
                                            AB is included
                                            between A and B.
                                            CB is included
                                            between C and B.
A                               B



March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   3
Postulate 21: Angle-Side-Angle
    If two angles and the included side of one
    triangle are congruent to corresponding parts
    of another triangle, then the triangles are
    congruent. (ASA)
      C                          Z



A                                B        X                           Y
                     ABC  XYZ
March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS       4
Example
Prove ADB  CBD
         D                 C
                                   Proof:

                              A ADB  CBD (Given)
                              S DB  DB (Ref.)
  A              B           A ABD  CDB (Given)
                                   ADB  CBD (ASA)


March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   5
Theorem 4.5 Angle-Angle-Side
   If two angles and a nonincluded side of
    one triangle are congruent to
    corresponding parts of another triangle,
    then the triangles are congruent.
         B                                                            E




     A                         C           F                              D

                 ABC  DEF
March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS           6
Example
Is ABC  ADC?
                     A

                                                     Yes,by AAS
                                    S

                 A
     B                                     D


           A         C

March 10, 2012           Geometry 4.4 Proving Triangles Congruent: ASA, AAS   7
Are these triangles congruent?
                                    Yes. Reason:
                                    AAS
                                    OR AAS.




March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   8
Are these triangles congruent?

                                    You could
                                    even use
                                    ASA.


March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   9
Overlapping Triangles
                      B                                C




                 A                                                   D




March 10, 2012       Geometry 4.4 Proving Triangles Congruent: ASA, AAS   10
Problem: Is ABD  DCA?
     B                   C
         85°       85°
           15°    15°
A                             D

                 YES
                 by
                 AAS
March 10, 2012      Geometry 4.4 Proving Triangles Congruent: ASA, AAS   11
Problem: Is ABD  DCA?
     B                   C                        B                            C
         85°       85°                                85°                85°

           15°    15°                                   15°         15°
A                             D             A                                      D

                 YES
                 by
                 AAS
March 10, 2012      Geometry 4.4 Proving Triangles Congruent: ASA, AAS                 12
Methods to show triangles
congruent:
SSS                     When trying to determine
                         if triangles are congruent,
SAS                     carefully study the given
                         information, mark the
ASA                     triangles if necessary,
                         then determine what
AAS                     pattern you have.


March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   13
                                                         T
Problem
                    S                 S                                  A
           B                  A


                                           S
    A                                  C
                                                           R

        A                                               A
                 What other congruence must be
                 given to show that ABC  RST
                 using the AAS postulate? BC  ST
March 10, 2012          Geometry 4.4 Proving Triangles Congruent: ASA, AAS   14
                                                       T
Problem Alternate Solution

                                                                       A
S          B


                                         S
    A                                C                                     A
                                                         R

         A                A               S
                 What other congruence must be
                 given to show that ABC  RST
                 using the AAS postulate? BA  SR
March 10, 2012        Geometry 4.4 Proving Triangles Congruent: ASA, AAS       15
                                                       T
Problem
                                                                           A
         B

                                                                           S
  A                              C      S
                                                                       A
                                                         R
       A S              A

                 What other congruence must be
                 given to show that ABC  RST
                 using the ASA postulate? AC  RT
March 10, 2012        Geometry 4.4 Proving Triangles Congruent: ASA, AAS       16
1. ASA post.
                 B                              F                             D




    A                                 C
                                                                          E

                                C  F

March 10, 2012       Geometry 4.4 Proving Triangles Congruent: ASA, AAS           17
2. AAS Post.
         A                 B                E                         D




C                 C  F                                                 F




March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS           18
3. SSS
                                    B                             D
                 C



                                                          F
             A                  E


                 CB  FE

March 10, 2012       Geometry 4.4 Proving Triangles Congruent: ASA, AAS   19
4. SAS Post.
                                  F                                       E
                 A




                                                    C                     D
                 B

                               A  D

March 10, 2012       Geometry 4.4 Proving Triangles Congruent: ASA, AAS       20
5.
                 W
                                           Yes
                                           AAS


      X                            Z

March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   21
6.               C                       A

                                               Yes
                                               SAS


     D                       R


March 10, 2012       Geometry 4.4 Proving Triangles Congruent: ASA, AAS   22
7.                                   The only other thing we
                     T               know for sure:
S
                                     SOT  LOF
                                     (Vertical Angles)
                 O
                                     But this spells…
                                     SSA
                           L
      F                              NOT CONGRUENT
March 10, 2012           Geometry 4.4 Proving Triangles Congruent: ASA, AAS   23
L                                  S
                                              Given: LA || SN, LR  NR
                 R
                                              Prove: LAR  NSR
    A
                                       N




    1. LA || SN                             1. Given
    2. L  N                              2. Alt. int s
    3. LR  NR                              3. Given
    4. LRA  NRS                          4. Vert s
    5. LAR  NSR                          5. ASA
March 10, 2012       Geometry 4.4 Proving Triangles Congruent: ASA, AAS   24
List four ways to show triangles are
congruent.
 SSS
 ASA
 SAS
 AAS
                 List one way that won’t work.
                                            SSA


March 10, 2012     Geometry 4.4 Proving Triangles Congruent: ASA, AAS   25
Important!
 To show two triangles congruent, at least
  one side must be known.
 Three angles (AAA) does not show
  congruence (but it is still useful).




March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   26
Homework
 Page 223
 Problems 8 – 19 all
 Check the odds when finished
 Tomorrow: review and practice all of the
  triangle congruence postulates.



March 10, 2012   Geometry 4.4 Proving Triangles Congruent: ASA, AAS   27
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