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Using HL Congruence Theorem to Prove Triangle Congruence

Learn how to use the HL Congruence Theorem to prove triangle congruence. Understand the conditions and reasoning behind this theorem.

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Using HL Congruence Theorem to Prove Triangle Congruence

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  1. 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.

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

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

  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 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. a.

  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.

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

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

  8. 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. yes; SSS Congruence Postulate, SAS Congruence Postulate, or HL Congruence Theorem yes; HL Congruence Theorem 2. no ANSWER ANSWER ANSWER 3.

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