Download 4.7 ASA and AAS - Nutley Public Schools

4.7 ASA and AAS
Objectives:
Apply ASA and AAS to construct triangles and to
solve problems.
Prove triangles congruent by using ASA and AAS.
Warm up
1. Given DEF and GHI, if D  G and E  H,
why is F  I?
Third s Thm.
• If two angles in one triangle are congruent to two
angles in another triangle, then the third pair of
angles must also congruent. This is called the Third
Angle Theorem.
• If ∠A≅∠D and ∠B≅∠E, then ∠C≅∠F.
Can someone explain what the
“included” angle means in SAS?
So explain what the
“included” side means in ASA?
• An included side is the common side of
two consecutive angles in a triangle. The
following theorem uses the idea of an
included side.
Name the included side between each pair of angles.
1. R and K
2. X and R
3. 8 and 9
4. 10 and 12
5. 5 and 1
6. 4 and 2
Example 1: Applying ASA Congruence
Determine if you can use ASA to prove the
triangles congruent. Explain.
Two congruent angle pairs are given, but the
included sides are not given as congruent. So, we
cannot use ASA.
Check It Out! Example 2
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.
You can use the Third Angles Theorem to prove
another congruence relationship based on ASA. This
theorem is Angle-Angle-Side (AAS).
State the postulate that you would use to prove
the triangles congruent. Name the congruent
triangles.
State the postulate that you would use to prove
the triangles congruent. Name the congruent
triangles.
Check It Out! Example 3
Prove the triangles congruent.
Given: JL bisects KLM, K  M
Prove: JKL  JML
A
A
S
S
A
A
✔ ✔✔
USE: AAS
Statements
Reasons
1. ÐK @ ÐM
1. Given A
2. JL bisects ÐKLM
2. Given
3. ÐKLJ @ ÐMLJ
4. JL @ JL
3. An angle bisector cuts an A
angle into 2 congruent angles
4. Reflexive Property S
5. DJKL @ DJML
5. AAS