1. 4-6 Warm Up Lesson Presentation Lesson Quiz
Triangle Congruence: ASA, AAS, and HL
Holt Geometry
4-5
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry

2. 1. What are sides AC and BC called? Side AB?
4-5
Warm Up
1. What are sides AC and BC called? Side AB?
2. Which side is in between A and C?
3. Given DEF and GHI, if D  G and E  H, why is F  I?
legs; hypotenuse AC
Third s Thm.

3. 4-5
Objectives
Apply ASA, AAS, and HL to construct triangles and to solve problems.
Prove triangles congruent by using ASA, AAS, and HL.

4. Can someone explain what the “included” angle means?
4-5
Can someone explain what the “included” angle means?

5. 4-5
An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

6. 4-5

7. Example 1: Applying ASA Congruence
4-5
Example 1: Applying ASA Congruence
Determine if you can use ASA to prove the triangles congruent. Explain.
Here period 3
Two congruent angle pairs are given, but the included sides are not given as congruent. So, we cannot use ASA.

8. 4-5
Check It Out! Example 1
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.

9. 4-5
You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS).

10. A A S S A A 4-5 Check It Out! Example 2 Prove the triangles congruent.
Given: JL bisects KLM, K  M
Prove: JKL  JML A A S S A A

11. USE: AAS A A S 4-5 1. Given 2. Given✔ ✔ ✔
USE: AAS
Statements
Reasons
1. Given A
2. Given
3. An angle bisector cuts an angle into 2 congruent angles A
4. Reflexive Property S
5. AAS

12. The bad word one… SSA or ASS…..
4-5
The bad word one…
SSA or ASS…..
We cannot use this to prove two triangles are congruent.
It’s a bad word, it’s bad news, and it doesn't’t work… so NEVER use it!

13. I think of this one as rASS
4-5
I think of this one as rASS

14. In a HL proof you need 3 things The hypotenuses The legs
4-5
In a HL proof you need 3 things
The hypotenuses
The legs
Tell us that the triangles have a right angle, thus making them right triangles.

15. Example 3A: Applying HL Congruence
4-5
Example 3A: Applying HL Congruence
Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know.
According to the diagram, the triangles are right triangles that share one leg.
It is given that the hypotenuses are congruent, therefore the triangles are congruent by HL.

16. Example 3B: Applying HL Congruence
4-5
Example 3B: Applying HL Congruence
This conclusion cannot be proved by HL. According to the diagram, the triangles are right triangles and one pair of legs is congruent. You do not know that one hypotenuse is congruent to the other.

17. 4-5
Check It Out! Example 3
Determine if you can use the HL Congruence Theorem to prove ABC  DCB. If not, tell what else you need to know.
Yes, so lets prove it!

18. USE: HL-RT H L RT 4-5 1. Given 2. Reflexive Property 3. Given✔ ✔ ✔
USE: HL-RT
Statements
Reasons
1. Given H
2. Reflexive Property L
3. Given
4. A right triangle has 1 right angle RT
5. HL

19. 4-5
Lesson Quiz: Part I
Identify the postulate or theorem that proves the triangles congruent. HL
ASA
SAS or SSS

20. 4-5 Lesson Quiz: Part II 4. Given: FAB  GED, ACB   DCE, AC  EC
Prove: ABC  EDC

21. Lesson Quiz: Part II Continued
4-5
Lesson Quiz: Part II Continued
5. ASA
5. ABC  EDC
4. Given
4. ACB  DCE; AC  EC
3.  Supp. Thm.
3. BAC  DEC
2. Def. of supp. s
2. BAC is a supp. of FAB; DEC is a supp. of GED.
1. Given
1. FAB  GED
Reasons
Statements