Download Proving Triangles are Congruent: ASA and AAS

4.5 Proving Triangles are
Congruent: ASA and AAS
Hubarth
Geometry
Angle-Side-Angle Congruence Postulate (ASA)
Words If two angles and the included side of one triangle
are congruent to two angles and the included side
of a second triangle, then the two triangles are congruent.
Symbols
If Angle A  X, and
Side AB  XY , and
Angle B  Y,
Then ABC  XYZ
Ex 1. Determine When to Use ASA Congruence
Based on the diagrams can you use the ASA Congruence Postulate to show
that the triangles are congruent? Explain your reasoning.
a.
B
O
M
W
A
b.
N
Solution
a. B  M , O  A, and BO  AM ,
So by ASA triangleBOW  AMN
b. A  C, ABB'  CBB' and AB  CB,
So, by ASA triangle ABB'  CBB'.
Angle-Angle-Side Congruence Theorem (AAS)
Words If two angles and a non-included side of one triangle
are congruent to two angles and the corresponding
non-included side of a second triangle, then the two
triangles are congruent.
Symbols
If Angle A  O, and
Angle N  W, and
Side BO  MA,
then BOW  MAN
Ex 2. Determine What Information Is Missing
What additional congruence is needed to show that JKL  NML
by the AAS.
K
N
L
J
M
Solution
Given: KL  ML
KLJ  MLN by vertical angles
To use AAS we will need to know J  N.
Ex 3. Decide Whether Triangles are Congruent
Does the diagram give enough information to show that the triangles are congruent?
If so, state the reason.
N
Q
a.
U
H
b.
E
Z
c.
G
F
J
M
W
P
X
Solution
a. EF  HJ
Given
E  J
Given
FGE  HGJ Vertical Angles
EFG  JHG AAS
b. MP  QN
Given
c. WZ  WZ
Reflexive Prop
NP  NP Reflexive Prop.
UZW  XWZ Alt. Int
No other information is
given to determine that
UWZ  XZW Alt Int
they are congruent.
UZW  XWZ
ASA
Ex 4. Prove Triangles are Congruent
A step in the Cat's Cradle string game creates
the triangles shown. Prove that ABD  EBC.
A
C
B
D
Solution
Given: BD  BC , AD EC
Prove: ABD  EBC
Statements
Reasons
1. BD  BC
1. Given
2. AD EC
3. D  C
2. Given
3. Alt. Int. angles
4. ABD  EBC
4. Vertical Angles
5. ABD  EBC
5. ASA
E
Practice
1.
Based on the diagram, can you use the ASA Congruence Postulate to show the triangles are congruent?
Explain.
E
D
B
A  D, B  E
AB  DE ,
ABC  DEF by ASA
F
A
C
2. Does the diagram give enough information to show that the triangles are congruent? If so state why.
B
A
Yes, AAS
D
C
3. Use the information in the diagram to prove that FGH  GJK.
F
G
H
G’
K
J
Statements
1. GFH  JG'K
Reasons
1. Given
2. GF  JG '
3. FGH  G'JK
4. FGH  G'JK
2. Given
3. Corresponding Angles
4. ASA