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4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
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Lesson Presentation
Exit Ticket
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Warm Up #2
1. What are sides AC and BC called? Side AB?
legs; hypotenuse
2. Which side is in between A and C?
AC
3. Given DEF and GHI, if D  G and E  H,
why is F  I?
Third s Thm.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Know: Solve It !
Given: 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐷 ∥ 𝐶𝐵
Prove: ∆ABC  ∆CDA
1. What do you need to find to solve the problem ?
at least three corresponding pairs of sides or angles
that we can prove to be congruent
y◦
2. What are the corresponding parts of the two triangles ?
D and B; DAC and BCA; CAB and ACD;
𝐴𝐶 and 𝐶𝐴; 𝐴𝐷 and 𝐶𝐵; 𝐴𝐵 and 𝐶𝐷
3. What word would you use to describe 𝑨𝑪 ? transversal
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Communicate: Connect Mathematical Ideas (1)(F)
Given: 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐷 ∥ 𝐶𝐵
Prove: ∆ABC  ∆CDA
4. What can you show about angles in the triangles that can
indicate congruency?
We can find congruent angles using alternate interior angles of
y◦
the transversal.
5. What do you know about a side or sides of the triangles that
can be used to show congruency?
The transversal is part of both triangles, so it is congruent to itself
by the Reflexive Property of Congruence.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Connect to Math
By the end of today’s lesson,
SWBAT
1. Apply ASA, AAS to construct triangles and to
solve problems.
2. Prove triangles congruent by using ASA, AAS,
and Applying CPCTC.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Vocabulary
included side
CPCTC
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Essential Understanding
You can prove that two triangles are congruent
without having to show that all corresponding
parts are congruent. In this lesson, you will
prove triangles congruent by using one pair of
corresponding sides and two pairs of
corresponding angles.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
An included side is the common side of two
consecutive angles in a polygon. The following
postulate uses the idea of an included side.
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Triangle Congruence: ASA, AAS, and Applying CPCTC
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 1: Applying ASA Congruence
Which two triangles are congruent by ASA ?
Explain.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 1: Applying ASA Congruence
Which two triangles are congruent by ASA ?
Explain.
In ∆SUV, 𝑈𝑉 included between U and
V and has a congruence marking.
In ∆NEO, 𝐸𝑂 included between E and
O and has a congruence marking.
In ∆ATW, 𝑇𝑊 included between T and
W but does not have congruence marking.
Since U ≌ E , 𝑈𝑉 ≅ 𝐸𝑂, and V ≌ O ; therefore, ∆SUV ≌ ∆NEO.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 2: Applying ASA Congruence
Determine if you can use ASA to prove the
triangles congruent. Explain.
Two congruent angle pairs are give, but the included sides are
not given as congruent. Therefore ASA cannot be used to prove
the triangles congruent.
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Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 3
Determine if you can use ASA to
prove NKL  LMN. Explain.
By the Alternate Interior Angles Theorem, KLN  MNL.
𝑁𝐿  𝐿𝑁 by the Reflexive Property. No other congruence
relationships can be determined, so ASA cannot be applied.
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Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 4: Angle-Side-Angle Congruence
Statements
Reasons
1. G  K; J  M
1. Given
2. H  L
2. Third s Thrm.
3. 𝐻𝐽  𝐿𝑀
3. Given
4. ∆GHJ  ∆KLM
4. ASA Steps 1, 3, 2
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Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 5: Writing a Proof
Recreation Members of a teen organization
are building a miniature golf course at your
town’s youth center. The design plan calls
for the first hole to have two congruent
triangular bumpers. Prove that the bumpers
on the first hole, shown at the right, meet
the conditions of the plan.
Given: 𝐴𝐵 ≅ 𝐷𝐸, A ≌ D, B and E are right angles
Prove: ∆ABC ≅ ∆DEF
Proof:
It is given that A ≌ D and 𝐴𝐵 ≅ 𝐷𝐸.
B ≌ E because all rights angles are
congruent. Hence, ∆ABC ≅ ∆DEF by ASA.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
You can use the Third Angles Theorem to prove another congruence
relationship based on ASA. This theorem is Angle-Angle-Side (AAS).
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Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 6: Writing a Proof Using AAS
Given: M ≌ K , 𝑊𝑀 ∥ 𝑅𝐾,
Prove: ∆WMR ≅ ∆RKW
Statements
Reasons
1. M  K
2. 𝑊𝑀 || 𝑅𝐾
1. Given
2. Given
3. MWR  KRW
3. Alt. Int. s Thm.
4. 𝑊𝑅  𝑊𝑅
5. ∆WMR  ∆RKW
4. Reflex. Prop. of 
5. AAS Steps 1, 3, 4
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Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 7: Writing a Proof Using AAS
Given: 𝐽𝐿 bisects KLM, K  M
Prove: JKL  JML
𝐽𝐿 bisects KLM
Given
KLJ ≌ MLJ
Def. of  bisector
K ≌ M
𝐽𝐿 ≌ 𝐽𝐿
Given
Reflexive Prop. of ≌
JKL  JML
AAS
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
CPCTC is an abbreviation for the
phrase “Corresponding Parts of
Congruent Triangles are Congruent.”
It can be used as a justification in a
proof after you have proven two
triangles congruent.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Remember !
SSS, SAS, ASA, and AAS use
corresponding parts to prove triangles
congruent. CPCTC uses congruent
triangles to prove corresponding parts
congruent.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 8: Writing a Proof Using CPCTC
Given: M ≌ K , 𝑊𝑀 ∥ 𝑅𝐾,
Prove: 𝑀𝑅 ≅ 𝐾𝑊
Statements
Reasons
1. M  K
1. Given
2. 𝑊𝑀 || 𝑅𝐾
3. MWR  KRW
4. 𝑊𝑅  𝑊𝑅
5. ∆WMR  ∆RKW
2. Given
3. Alt. Int. s Thm.
4. Reflex. Prop. of 
6. 𝑀𝑅 ≅ 𝐾𝑊
6. CPCTC
5. AAS Steps 1, 3, 4
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Triangle Congruence: ASA, AAS, and Applying CPCTC
Example 9: Landscaping Application
A landscape architect sets up the
triangles shown in the figure to find
the distance 𝑱𝑲 across a pond.
What is 𝑱𝑲?
One angle pair is congruent, because they are vertical angles.
Two pairs of sides are congruent, because their lengths are equal.
Therefore the two triangles are congruent by SAS. By CPCTC, the
third side pair is congruent, so 𝐽𝐾 = 41 ft.
4-3
Triangle Congruence: ASA, AAS, and Applying CPCTC
Exit Ticket
Identify the postulate or theorem that proves the
triangles congruent.
2. Given: FAB  GED, ABC   DCE, AC  EC
Prove: BC  DC