1. In the diagram, you are given that ∆JGH and ∆HKJ are right triangles.
Example 1
Determine When To Use HL
Is it possible to show that ∆JGH  ∆HKJ using the HL Congruence Theorem? Explain your reasoning.
SOLUTION
In the diagram, you are given that ∆JGH and ∆HKJ are right triangles.
By the Reflexive Property, you know JH  JH (hypotenuse) and you are given that JG  HK (leg). You can use the HL Congruence Theorem to show that ∆JGH  ∆HKJ. 1

2. Use the diagram to prove that ∆PRQ  ∆PRS.
Example 2
Use the HL Congruence Theorem
Use the diagram to prove that ∆PRQ  ∆PRS.
SOLUTION SQ  PR PS
PQ 
∆PRQ  ∆PRS
Statements
Reasons
Given 1. SQ  PR 2.
PRQ and PRS are right . S
lines form right angles.  2

3. ∆PRQ and ∆PRS are right triangles. Definition of right triangle
Example 2
Use the HL Congruence Theorem
Statements
Reasons 3.
∆PRQ and ∆PRS are right triangles.
Definition of right triangle 4.
Given PS
PQ 
Reflexive Prop. of Congruence 5. PR
PR  6.
∆PRQ  ∆PRS
HL Congruence Theorem 3

4. Example 3
Decide Whether Triangles are Congruent
Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. a. b.
SOLUTION a.
From the diagram, you know that BAC  DAC, B  D, and BC  DC. You can use the AAS Congruence Theorem to show that ∆BAC  ∆DAC. 4

5. Example 3
Decide Whether Triangles are Congruent b.
From the diagram, you know that FG  HG, EG  EG, and EFG  EHG. Because the congruent angles are not included between the congruent sides, you cannot show that ∆FGE  ∆HGE. 5

6. Use the information in the diagram to prove that ∆RST  ∆UVW.
Example 4
Prove Triangles are Congruent
Use the information in the diagram to prove that ∆RST  ∆UVW.
SOLUTION
Statements
Reasons 1.
S  V
Given
ST  VW 2.
Given 3.
∆UVW is equilateral.
Definition of equilateral triangle
V  W 4.
Equilateral triangles are equiangular. 6

7. Transitive Prop. of Congruence
Example 4
Prove Triangles are Congruent
Statements
Reasons 5.
T  V
Given
T  W 6.
Transitive Prop. of Congruence 7.
∆RST  ∆UVW
ASA Congruence Postulate 7

8. yes; HL Congruence Theorem
Checkpoint
Decide Whether Triangles are Congruent
Does the diagram give enough information to show that the triangles are congruent? If so, state the postulate or theorem you would use. 1.
ANSWER
yes; HL Congruence Theorem 2.
ANSWER no
ANSWER
yes; SSS Congruence Postulate, SAS Congruence Postulate, or HL Congruence Theorem 3.