4.4 Proving Triangles are Congruent: ASA and AAS
Download
Report
Transcript 4.4 Proving Triangles are Congruent: ASA and AAS
4.4 Proving Triangles are
Congruent: ASA and
AAS
Geometry
Ms. Reser
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.
Assignment
• 4.4 pp. 223-225 #1-22 all
• Quiz after this section
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 F, C
D, BA 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
and one side of ∆ABC are
congruent to two angles and
one side of ∆DEF. By the
Third Angles Theorem, the
third angles are also
congruent. That is, B E.
Notice that BA is the side
C
included between B and
A, and EF 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.