Download ASA and AAS Triangle Congruence

ASA and AAS Triangle
Congruence
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: August 23, 2016
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
C HAPTER
Chapter 1. ASA and AAS Triangle Congruence
1
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.
Angle-Side-Angle Postulate and Angle-Angle-Side Theorem
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
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?
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/136751
1
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Examples
Example 1
Can you prove that the following triangles are congruent? Why or why not?
We cannot show the triangles are congruent because KLand ST are not corresponding, even though they are congruent. To determine if KLand ST are corresponding, look at the angles around them, 6 Kand 6 Land 6 Sand 6 T . 6 Khas
one arc and 6 Lis unmarked. 6 Shas two arcs and 6 T is unmarked. In order to use AAS, 6 Sneeds to be congruent to
6 K.
Example 2
Write a 2-column proof.
Given: AB||ED, 6 C ∼
= 6 F, AB ∼
= ED
Prove: AF ∼
= CD
TABLE 1.1:
Statement
1. AB||ED, 6 C ∼
= 6 F, AB ∼
= ED
6
2. 6 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)
Example 3
What information do you need to prove that these two triangles are congruent using the ASA Postulate, AB ∼
=
UT , AC ∼
= UV , BC ∼
= TV , or 6 B ∼
= 6 T?
2
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Chapter 1. ASA and AAS Triangle Congruence
For ASA, we need the side between the two given angles, which is AC and UV . The answer is AC ∼
= UV .
Example 4
Write a 2-column proof.
∼ 6 E, AC =
∼ AE
Given: 6 C =
Prove: 4ACF ∼
= 4AEB
TABLE 1.2:
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 5
What information do you need to prove that these two triangles are congruent using ASA? AAS?
For ASA, we need the angles on the other side of EF and QR. 6 F ∼
=6 Q
For AAS, we would need the other angle. 6 G ∼
=6 P
3
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Review
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?
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
6
8. What would be your reason for C = A?
For questions 9-13, use the picture and the given information.
Given: LP||NO, LP ∼
= NO
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9.
10.
11.
12.
13.
Chapter 1. ASA and AAS Triangle Congruence
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 1.3:
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
16. ASA
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17. AAS
Review (Answers)
To see the Review answers, open this PDF file and look for section 4.8.
Resources
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/1303
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/1304
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/1317
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