1. Triangle Congruence by SSS and SAS
GEOMETRY LESSON 4-2
1. In VGB, which sides include B?
2. In STN, which angle is included between NS and TN?
3. Which triangles can you prove congruent?
Tell whether you would use the SSS or SAS Postulate.
4. What other information do you need to prove
DWO DWG?
5. Can you prove SED BUT from the information given? Explain.
BG and BV N
APB XPY; SAS
If you know DO DG, the triangles are by SSS; if you know DWO DWG, they are by SAS.
No; corresponding angles are not between corresponding sides.
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2. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
(For help, go to Lesson 4-2.)
In JHK, which side is included between the given pair of angles?
1.  J and  H 2.  H and  K
In NLM, which angle is included between the given pair of sides?
3. LN and LM 4. NM and LN
Give a reason to justify each statement.
5. PR PR 6.  A  D L N
By the Reflexive Property of Congruence, a segment is congruent to itself
Third Angles Theorem
Check Skills You’ll Need
4-3

5. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.
4-3

6. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
4-3

7. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS).
4-3

8. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
4-3

9. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
Using ASA
Suppose that F is congruent to C and I is not congruent to C. Name the triangles that are congruent by the ASA Postulate.
The diagram shows N A D and FN CA GD.
If F C, then F C G
Therefore, FNI CAT GDO by ASA.
Quick Check
4-3

10. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
Writing a proof using ASA
Write a paragraph proof.
Given: A B, AP BP
Prove: APX BPY
It is given that A B and AP BP.
APX BPY by the Vertical Angles Theorem.
Because two pairs of corresponding angles and their included sides are congruent, APX BPY by ASA.
Quick Check
4-3

11. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
Planning a Proof using AAS
Write a Plan for Proof that uses AAS.
Given: B D, AB || CD
Prove: ABC CDA
Because AB || CD, BAC DCA by the Alternate Interior Angles Theorem.
Then ABC CDA if a pair of corresponding sides are congruent.
By the Reflexive Property, AC AC so ABC CDA by AAS.
Quick Check
4-3

12. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
Writing a proof using AAS
Write a two-column proof that uses AAS.
Given: B D, AB || CD
Prove: ABC CDA
Statements Reasons
1. B D, AB || CD 1. Given
2. BAC & DCA are AIA 2. Definition of Alternate Interior Angle.
3. BAC DCA 3. Alternate Interior Angle Theorem .
4. AC CA 4. Reflexive Property of Congruence
5. ABC CDA 5. AAS Theorem
Quick Check
4-3

13. Triangle Congruence by ASA and AAS
GEOMETRY LESSON 4-3
1. Which side is included between R and F in FTR?
2. Which angles in STU include US?
Tell whether you can prove the triangles congruent by ASA or AAS. If you can, state a triangle congruence and the postulate or theorem you used. If not, write not possible. RF
S and U
GHI PQR AAS
not possible
ABX ACX AAS
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