Download ASA and AAS Triangle Congruence

ASA and AAS Triangle
Congruence
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: November 21, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. ASA and AAS Triangle Congruence
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ASA and AAS Triangle
Congruence
Here you’ll learn how to prove that triangles are congruent given information only about two of their angles and one
of their sides.
What if you were given two triangles and provided with only the measure of two of their angles and one of their side
lengths? How could you determine if the two triangles were congruent? After completing this Concept, you’ll be
able to use the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) shortcuts to prove triangle congruency.
Watch This
MEDIA
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CK-12 ASA and AAS Triangle Congruence
Watch the portions of the following two videos that deal with ASA and AAS triangle congruence.
MEDIA
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James Sousa:Introduction toCongruent Triangles
MEDIA
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James Sousa:DeterminingIf Two Triangles are Congruent
Finally, watch this video.
MEDIA
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James Sousa:Example2: Prove Two Triangles are Congruent
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Guidance
If two angles and one side in one triangle are congruent to the corresponding two angles and one side in another
triangle, then the two triangles are congruent. This idea encompasses two triangle congruence shortcuts: AngleSide-Angle and Angle-Angle-Side.
Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side in one triangle are congruent
to two angles and the included side in another triangle, then the two triangles are congruent.
Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side in one triangle are congruent to two angles and the corresponding non-included side in another triangle, then the triangles are congruent.
The placement of the word Side is important because it indicates where the side that you are given is in relation to
the angles. The pictures below help to show the difference between the two shortcuts.
ASA
AAS
Example A
What information do you need to prove that these two triangles are congruent using the ASA Postulate?
a) AB ∼
= UT
b) AC ∼
= UV
c) BC ∼
= TV
d) 6 B ∼
=6 T
For ASA, we need the side between the two given angles, which is AC and UV . The answer is b.
Example B
Write a 2-column proof.
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Concept 1. ASA and AAS Triangle Congruence
Given: 6 C ∼
= 6 E, AC ∼
= AE
Prove: 4ACF ∼
= 4AEB
TABLE 1.1:
Statement
1. 6 C ∼
= 6 E, AC ∼
= AE
∼
6
6
2. A = A
3. 4ACF ∼
= 4AEB
Reason
1. Given
2. Reflexive PoC
3. ASA
Example C
What information do you need to prove that these two triangles are congruent using:
a) ASA?
b) AAS?
Solution:
a) For ASA, we need the angles on the other side of EF and QR. 6 F ∼
=6 Q
b) For AAS, we would need the other angle. 6 G ∼
=6 P
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CK-12 ASA and AAS Triangle Congruence
Guided Practice
1. Can you prove that the following triangles are congruent? Why or why not?
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2. Write a 2-column proof.
Given: BD is an angle bisector of 6 CDA, 6 C ∼
=6 A
Prove: 4CBD ∼
= 6 ABD
3. Write a 2-column proof.
Given: AB||ED, 6 C ∼
= 6 F, AB ∼
= ED
Prove: AF ∼
= CD
Answers:
1. We cannot show the triangles are congruent because KL and ST are not corresponding, even though they are
congruent. To determine if KL and ST are corresponding, look at the angles around them, 6 K and 6 L and 6 S and
6 T . 6 K has one arc and 6 L is unmarked. 6 S has two arcs and 6 T is unmarked. In order to use AAS, 6 S needs to be
congruent to 6 K.
2.
TABLE 1.2:
Statement
1. BD is an angle bisector of 6 CDA, 6 C ∼
=6 A
∼
6
6
2. CDB = ADB
3. DB ∼
= DB
4. 4CBD ∼
= 4ABD
3.
4
Reason
1. Given
2. Definition of an Angle Bisector
3. Reflexive PoC
4. AAS
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Concept 1. ASA and AAS Triangle Congruence
TABLE 1.3:
Statement
1. AB||ED, 6 C ∼
= 6 F, AB ∼
= ED
∼
6
6
2. ABE = DEB
3. 4ABF ∼
= 4DEC
4. AF ∼
CD
=
Reason
1. Given
2. Alternate Interior Angles Theorem
3. ASA
4. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
Practice
For questions 1-3, determine if the triangles are congruent. If they are, write the congruence statement and which
congruence postulate or theorem you used.
1.
2.
3.
For questions 4-8, use the picture and the given information below.
Given: DB ⊥ AC, DB is the angle bisector of 6 CDA
4. From DB ⊥ AC, which angles are congruent and why?
5. Because DB is the angle bisector of 6 CDA, what two angles are congruent?
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6. From looking at the picture, what additional piece of information are you given? Is this enough to prove the
two triangles are congruent?
7. Write a 2-column proof to prove 4CDB ∼
= 4ADB, using #4-6.
6
8. What would be your reason for 6 C ∼
A?
=
For questions 9-13, use the picture and the given information.
Given: LP||NO, LP ∼
= NO
9.
10.
11.
12.
13.
From LP||NO, which angles are congruent and why?
From looking at the picture, what additional piece of information can you conclude?
Write a 2-column proof to prove 4LMP ∼
= 4OMN.
What would be your reason for LM ∼
= MO?
Fill in the blanks for the proof below. Use the given from above. Prove: M is the midpoint of PN.
TABLE 1.4:
Statement
1. LP||NO, LP ∼
= NO
2.
3.
4. LM ∼
= MO
5. M is the midpoint of PN.
Reason
1. Given
2. Alternate Interior Angles
3. ASA
4.
5.
Determine the additional piece of information needed to show the two triangles are congruent by the given postulate.
14. AAS
15. ASA
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Concept 1. ASA and AAS Triangle Congruence
16. ASA
17. AAS
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