5.3 Proving Triangles are Congruent: ASA and AAS
[Pages:4]5.3 Proving Triangles are Congruent:
ASA and AAS
Goal Show triangles are congruent using ASA and AAS.
POSTULATE 14: 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 aA c aD, and Side A**C** c *D**F*, and Angle aC c aF,
then TABC c T DEF .
E
B
D
F
A
C
Follow-Up
Use TTGL shown. Complete the table.
Angles
Included Side
T
aT and aG
T&G*
aG and aL
G&L*
aT and aL
T&L
G
L
Draw any TABC in the space below. Complete the table.
Angles
Non-Included Sides
B
aA and aB
A&C* and B&C*
aB and aC
A&B* and A&C*
A
aA and aC
A&B* and B&C*
C
114 Geometry, Concepts and Skills Notetaking Guide ? Chapter 5
Example 1 Determine When to Use ASA
Based on the diagrams, can you use the ASA Congruence Postulate to show that the triangles are congruent? Explain.
a.
B
D
b. T
Z
A
C
R
SX
Y
Solution
a. In TABC, the included side between the marked angles is B&C* . In TDCB, the included side is B&C* . These sides are congruent by the Reflexive Property. So, with aACB c a DBC , aABC c a DCB , and B&C* c B&C* , you can use the ASA Congruence Postulate to show that TABC c TDCB.
b. In TRST, the included side is R&S* . In TYXZ, the included side is X&Y* . Since these sides are not marked congruent, you cannot use the ASA Congruence Postulate.
THEOREM 5.1: 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
E
If Angle aA c aD, and
Angle aC c aF, and
BD
F
Side *B*C** c *E*F**,
then TABC c T DEF .
A
C
Follow-Up
In Example 1, part (b), can you use the AAS Congruence Theorem to show that the triangles are congruent?
Yes
Lesson 5.3 ? Geometry, Concepts and Skills Notetaking Guide 115
Example 2 Determine What Information is Missing
What additional congruence is needed K
N
to show that TJKL c TNML by the
AAS Congruence Theorem?
L
Solution
J
M
You are given *K**L* c M&L*. You know aKLJ c aMLN because they
are vertical angles. The angles that make *K**L* and M&L* the non-included sides are a J and a N . So, to use the AAS
Congruence Theorem you need the congruence aJ c aN .
Example 3 Decide Whether Triangles are Congruent
Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use.
a. E
H
b.
N
Q c.
U
Z 1
2
G
F
J
M
P
4
3
W
X
Solution a. Use the AAS Congruence Theorem.
Side *E*F** c J**H** is given. Angle aE c aJ is given. Angle aFGE c aHGJ since vertical angles are congruent.
b. You only know that M***P* c Q**N** and N**P** c N**P**. You cannot conclude that the triangles are congruent.
c. Use the ASA Congruence Postulate. Angle a1 c a3 since alternate interior angles are congruent. Angle a2 c a4 since alternate interior angles are congruent. Side W***Z** c W***Z** by the Reflexive Property of Congruence.
116 Geometry, Concepts and Skills Notetaking Guide ? Chapter 5
Example 4 Prove Triangles are Congruent
Use the information given in the
A
diagram to prove TABD c TEBC.
C
Solution Given A**D** *E*C**, B**D** c *B*C** Prove TABD c TEBC
B
D
E
Statements
Reasons
1. B**D** c *B*C** 2. A**D** E**C** 3. aD c aC 4. aABD c aEBC 5. TABD c TEBC
1. Given 2. Given 3. Alternate Interior Angles Theorem 4. Vertical Angles Theorem 5. ASA Congruence Postulate
Follow-Up
In Example 4, how can you use the AAS Congruence Theorem to prove that the triangles are congruent? Replace Statement 3 with aA c aE and Reason 5 with AAS Congruence Theorem.
Checkpoint Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use.
1. A
1 4
B 2.
G
E
J 3. K
2
3
D
C
P
H
D
F
N L
M
Yes; ASA
No
Yes; AAS
Lesson 5.3 ? Geometry, Concepts and Skills Notetaking Guide 117
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