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Transcript
4.5 Proving Triangles are
Congruent: ASA and
AAS
Geometry
Objectives:
1. Prove that triangles are congruent
using the ASA Congruence Postulate
and the AAS Congruence Theorem
2. Use congruence postulates and
theorems in real-life problems.
Postulate 21: Angle-Side-Angle
(ASA) Congruence Postulate
• If two angles and the
B
included side of one
triangle are
congruent to two
angles and the
C
included side of a
second triangle, then
the triangles are
congruent.
A
E
F
D
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
• If two angles and a
B
non-included side of
one triangle are
congruent to two
angles and the
corresponding non- C
included side of a
second triangle, then
the triangles are
congruent.
A
E
F
D
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
Given: A  D, C
 F, BC  EF
Prove: ∆ABC  ∆DEF
B
A
E
C
F
D
Theorem 4.5: Angle-Angle-Side
(AAS) Congruence Theorem
You are given that two angles of
∆ABC are congruent to two
angles of ∆DEF. By the Third
Angles Theorem, the third
angles are also congruent.
That is, B  E. Notice that
BC is the side included
between B and C, and EF C
is the side included between
E and F. You can apply
the ASA Congruence
Postulate to conclude that
∆ABC  ∆DEF.
B
A
E
F
D
Ex. 1 Developing Proof
Is it possible to prove
the triangles are
congruent? If so,
state the postulate or
theorem you would
use. Explain your
reasoning.
H
E
G
F
J
Ex. 1 Developing Proof
A. In addition to the angles
and segments that are
marked, EGF JGH
by the Vertical Angles
Theorem. Two pairs of
corresponding angles
and one pair of
corresponding sides are
congruent. You can use
the AAS Congruence
Theorem to prove that
∆EFG  ∆JHG.
H
E
G
F
J
Ex. 1 Developing Proof
Is it possible to prove
the triangles are
congruent? If so,
state the postulate or
theorem you would
use. Explain your
reasoning.
N
M
Q
P
Ex. 1 Developing Proof
B. In addition to the
congruent segments
that are marked, NP
 NP. Two pairs of
corresponding sides
are congruent. This
is not enough
information to prove
the triangles are
congruent.
N
M
Q
P
Ex. 1 Developing Proof
Is it possible to prove
the triangles are
congruent? If so,
state the postulate or
theorem you would
use. Explain your
reasoning.
UZ ║WX AND UW
║WX.
U
1
2
W
3
4
X
Z
Ex. 1 Developing Proof
The two pairs of
parallel sides can be
used to show 1 
3 and 2  4.
Because the included
side WZ is congruent
to itself, ∆WUZ 
∆ZXW by the ASA
Congruence
Postulate.
U
1
2
W
3
4
X
Z
Ex. 2 Proving Triangles are
Congruent
Given: AD ║EC, BD  BC
Prove: ∆ABD  ∆EBC
Plan for proof: Notice that
ABD and EBC are
congruent. You are
given that BD  BC
. Use the fact that AD ║EC
to identify a pair of
congruent angles.
C
A
B
D
E
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1.
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
4. Vertical Angles
Theorem
C
A
Proof:
B
D
Statements:
1. BD  BC
2. AD ║ EC
3. D  C
4. ABD  EBC
5. ∆ABD  ∆EBC
E
Reasons:
1. Given
2. Given
3. Alternate Interior
Angles
4. Vertical Angles
Theorem
5. ASA Congruence
Theorem
Note:
• You can often use more than one method
to prove a statement. In Example 2, you
can use the parallel segments to show
that D  C and A  E. Then you
can use the AAS Congruence Theorem to
prove that the triangles are congruent.