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Lesson 4-5
Proving Congruence:
ASA and AAS
Transparency 4-5
5-Minute Check on Lesson 4-4
Determine which postulate can be used to prove that the triangles
are congruent. If it is not possible to prove they are congruent,
write not possible.
1.
2.
4.
5.
6.
3.
If AB  RS and BC  ST, what additional
congruence statement would be necessary to prove ABC  RST by
the SAS postulate?
D B  S
C A  T
A  R
A
B C  T
Standardized Test Practice:
Transparency 4-5
5-Minute Check on Lesson 4-4
Determine which postulate can be used to prove that the triangles
are congruent. If it is not possible to prove they are congruent,
write not possible.
1.
2.
SAS
4.
SSS
SSS
5.
not possible
6.
3.
SSS
If AB  RS and BC  ST, what additional
congruence statement would be necessary to prove ABC  RST by
the SAS postulate?
D B  S
C A  T
A  R
A
B C  T
Standardized Test Practice:
Objectives
• Use the ASA Postulate to test for triangle
congruence
• Use the AAS Theorem to test for triangle
congruence
Vocabulary
• Included side – the side in common between
two angles (end points are the vertexes)
Postulates and Theorems
• Angle-Side-Angle (ASA) Postulate: If
two angles and the included side of one
triangle are congruent to two angles and the
included side of another triangle, then the
triangles are congruent.
• Angle-Angle-Side (AAS) Theorem: If
two angles and a non-included side of one
triangle are congruent to the corresponding
two angles and side of another triangle,
then the triangles are congruent.
Angle – Side – Angle (ASA)
Given: AC = CD
A  D
Prove:
ABC 
DEC
Statements
Reasons
A  D
Given in problem
AC = CD (included side)
Given
ACB  DCE
ABC 
DEC
Vertical Angles Theorem
ASA Postulate
Write a paragraph proof.
Given: L is the midpoint of
Prove: WRL EDL
Proof:
because alternate interior angles are
congruent. By the Midpoint Theorem,
Since vertical angles are congruent,
WRL EDL by ASA.
Write a flow proof.
Given:
Prove:
Proof:
Write a flow proof.
Given:
Prove:
Proof:
STANCES When Ms. Gomez puts her hands on her hips,
she forms two triangles with
her upper body and arms.
Suppose her arm lengths AB
and DE measure 9 inches, and
AC and EF measure 11 inches.
Also suppose that you are
given that
Determine
whether ABC EDF.
Justify your answer.
Explore We are given measurements of two sides of
each triangle. We need to determine whether the
two triangles are congruent.
Plan
Since
Likewise,
We are given
Check each possibility using the five methods
you know.
Solve
We are given information about three sides. Since
all three pairs of corresponding sides of the
triangles are congruent, ABC EDF by SSS.
Examine You can measure each angle in ABC and EDF
to verify that
Answer: ABC EDF by SSS
The curtain decorating the window forms 2 triangles
at the top. B is the midpoint of AC. AE = 13 inches and
CD = 13 inches. BE and BD each use the same amount
of material, 17 inches. Determine whether ABE  CBD
Justify your answer.
Answer: ABE CBD by SSS
Summary & Homework
• Summary:
– If two pairs of corresponding angles and the
included sides of two triangles are congruent,
then the triangles are congruent (ASA)
– If two pairs of corresponding angles and a pair of
corresponding non-included sides of two
triangles are congruent, then the triangles are
congruent (AAS)
• Homework:
– pg 211 - 212: 15-20 all in two-column proof format