Pearson Unit 1 Topic 4: Congruent Triangles 4-6: Congruence in Right Triangles Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (6)(B) Prove two triangles are congruent by applying the Side-Angle-Side, Angle- Side-Angle, Side-Side-Side, Angle- Angle-Side, and Hypotenuse-Leg congruence conditions. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Vocabulary for right triangles: Hypotenuse—the side opposite the right angle. Legs—the 2 sides that form the right angle. hypotenuse leg leg

Don’t think that all right triangles will use HL to prove congruence! But only right triangles can use HL.

Why does this congruence method only have two letters (HL), while all of the other methods (SSS, SAS, ASA, AAS) have three letters? Remember that only right triangles have a hypotenuse. Therefore, the H implies both a right angle in each triangle and that the hypotenuses are congruent. The L could be either the pair of short legs, or the pair of long legs that are congruent.

Example: 1 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.

Example: 2 Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 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.

Example 3: Yes—the hypotenuses are congruent and one pair of legs are congruent. ΔLMP  ΔOMN by HL.

Example 4: 3y – 2 = ½ y + 4 3x + 1 = 6x - 8 3/2 y – 2 = 4 1 = 3x – 8

Example 5 You need to know that GHI and JGI are right angles.

Example: 6 STATEMENTS REASONS 1. 2. 3. 4. 5.

Example 7: 1. PRS and RPQ are right angles 1. Given 2. SP  QR Statement Reason 1. PRS and RPQ are right angles 1. Given 2. SP  QR 2. Given 3. PR  RP 3. Reflexive Prop of Congruence 4. PRS  RPQ 4. HL