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4.5 – Prove Triangles Congruent by ASA and AAS
Angle-Side-Angle (ASA) Congruence Postulate
E
A
4cm
A  C
AB  CD
B  D
F
B
C
4cm
D
If two angles and the ______________
included
side of
congruent to two angles
one triangle are _____________
included
and the ____________
side of a second triangle,
then the two triangles are ________________.
congruent
Angle-Angle-Side (AAS) Congruence Theorem
E
A
F
B
C
D
OR
Angle-Angle-Side (AAS) Congruence Theorem
E
A
F
B
C
D
nonincluded
If two angles and a _____________________
congruent
side of one triangle are _______________
to two
angles and the corresponding
nonincluded
____________________
side of a second
triangle, then the two triangles are
congruent
________________.
Does AAA work?
Does AAA work?
NO!
Does ASS work?
No!
There is no ASS in geometry!
Given: AB CD
AC BD
Prove: ∆ABC  ∆DCB
Statements
Reasons
1. AB CD
1.
2. BCD  CBA
3. AC BD
2.
3.
4.
5.
6.
4. BCA  CBD
5. BC  BC
6. ∆ABC  ∆DCB
Given
alternate interior angles
Given
alternate interior angles
Reflexive Prop.
ASA
Given: AE bisects DC at B
A  E
Prove: ABD  EBC
Statements
Reasons
1. AE bisects DC at B 1. Given
2. Def. of segment bisector
2. DB  BC
3. A  E
4. ABD  EBC
5. ∆ABD  ∆EBC
3. Given
4. vertical angles
5. AAS
Decide whether the triangles are congruent. Explain your reasoning.
Yes, ASA
Decide whether the triangles are congruent. Explain your reasoning.
Yes, AAS
Decide whether the triangles are congruent. Explain your reasoning.
Yes, SAS
Decide whether the triangles are congruent. Explain your reasoning.
No,
ASS doesn’t work!
Decide whether the triangles are congruent. Explain your reasoning.
Yes, AAS