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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
D                 C
Proof:

S DB  DB (Ref.)
A              B           A ABD  CDB (Given)

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
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
7

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