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

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

∆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

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

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

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

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

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.