Hypotenuse-Leg Congruence Theorem: HL Section: 5.4

Objective: Use the HL Congruence Theorem and summarize congruence postulates and theorems.

Key Vocabulary - Review Hypotenuse Leg of a right triangle

Theorems 5.2 Hypotenuse-Leg Congruence Theorem (HL)

Right Triangles Right Triangles: side across from right angle is the hypotenuse, the remaining two are legs. The longest side of a right triangle is the hypotehuse.

Congruent Right Triangles 3 5 There is one additional congruence rule that applies only to right triangles The two right triangles have two congruent sides, the hypotenuse and one leg Is this enough information to show that the two triangles are congruent? Remember, SSA is not a congruence rule Since ∆TEA and ∆CUP are both right triangles, we can use the Pythagorean Theorem to find the length of the other leg ∆TEA  ∆CUP by SSS (or SAS) 4 4 U C P 3 5

Hypotenuse-Leg Congruence Theorem (HL) Words; If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.. Symbols; A D If ∆ABC and ∆DEF are right triangles, and H L B C F then ∆ABC  ∆DEF E

Examples OR A M hypotenuse leg hypotenuse P N C B leg A M hypotenuse

HL Theorem To use the HL Theorem, you must show that three conditions are met. There are two right triangles. The triangles have congruent hypotenuses. There is one pair of congruent legs.

Example 1 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. 10

Your Turn: Tell whether the pair of triangles is congruent or not and why. ANSWER Yes; HL ≅ Thm.

Your Turn: Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. FGH, HJK ANSWER HL ≅ Thm.

Review: Triangle Congruence Shortcuts

Triangle Congruence Shortcut Practice

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. 15

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. 16

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.

Is there enough given information to prove the triangles congruent Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem. 1. ABE, CBD ANSWER SAS ≅ Post.

State a third congruence that would allow you to prove ΔRST ≅ ΔXYZ by the SAS Congruence postulate. 2. ST ≅ YZ, RS≅ XY ANSWER ∠S ≅ ∠Y

Joke Time What do you call a man who lives in an envelope? Bill. What dog can't bark? A hot dog. What do you call a monkey on a mine field? A Ba-Boom.

Assignment Pg 260 – 263:#1 – 39 odd