Download 4.4 Proving Triangles are Congruent: ASA and AAS

4.1 – 4.3 Triangle
Congruency
Geometry
SWBAT:
1. Recognize congruent triangles
2. Prove that triangles are congruent using
the ASA Congruence Postulate, SSS
Congruence Postulate, SAS Congruence
Postulate and the AAS Congruence
Theorem
3. Use congruence postulates and
theorems in real-life problems.
Congruent
• Means that corresponding parts are
congruent,
• Matching sides and angles will be
congruent
B
A
C
Y
X
Z
Naming
• ORDER MATTERS!!!!
Example 1
R
• If two triangles are congruent…
– Name all congruent angles
X
S
– Name all congruent sides
T
Y
Z
Reminder…
• If two angles of one triangle are
congruent to two angles of another
triangle then the 3rd angles are congruent
Keep in mind
• You can flip, turn or slide congruent
triangles and they will maintain
congruency!!
SSS
• If sides of one triangle are congruent to
sides of a second triangle then the two
triangles are congruent.
Example 1
• Given STU with S(0, 5), T(0,0), and U(-2,
0) and XYZ with X (4, 8), Y (4, 5), and Z
(6, 3), determine if STU  XYZ
Included angles
SAS
• If two sides and the included angle of one
triangle are congruent to two sides and
included angle of another triangle then
the two triangles are congruent.
Example 2
• Write a proof
– Given: X is the midpoint of BD
X is the midpoint of AC
D
- Prove: DXC  BXA
A
X
C
B
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
• If asked to show that a pair of
corresponding parts of two triangles are
congruent you often prove the triangles
are congruent the by definition of
congruent triangles the parts are
congruent (CPCTC)
Example 3
• Given:
VR  RS
UT  SU
SR  TU
• Prove: VR  TU
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
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, MP
 NP. Two pairs of
corresponding sides
are congruent. This
is not enough
information to prove
the triangles are
congruent.
N
M
Q
P
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.