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Hypotenuse Leg Congruence Theorem Extension
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a and b are the legs of the right triangle a
Legs of a Right Triangle Hypotenuse of a Right Triangle Hypotenuse-Leg (HL) Congruence Theorem Any two right triangles with a congruent hypotenuse and a corresponding congruent leg are congruent triangles c a and b are the legs of the right triangle a b c c is the hypotenuse of a right triangle a b
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Explain why these two triangles are congruent using the hypotenuse leg congruence theorem.
Corresponding leg = 𝐵𝐶 ≅ 𝐵𝐶 The hypotenuse of each triangle are congruent and each triangle has a corresponding side that is congruent so therefore they are congruent by HL.
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Example 3: Using the given postulate, tell which parts of the pair of triangles should be shown congruent. 𝑅𝑃 ≅ 𝑄𝑆
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Try the you try before looking at the answer!
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You Try! 1. Why can these two triangles be considered congruent?
The hypotenuse and legs are congruent, therefore the two triangles are congruent.
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You Try! 2. Is it possible to show that JGH HJK?
Yes, due to the reflexive property the hypotenuse 𝐽𝐻 ≅ 𝐽𝐻 , each triangle also has a corresponding leg that is congruent ( 𝐽𝐺 ≅ 𝐻𝐾 ).
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Example 4: 𝐷𝐸 ⊥ 𝐹𝐶 ∠𝐷𝐸𝐹 & ∠𝐷𝐸𝐶 are right angles
𝐷𝐸 ⊥ 𝐹𝐶 ∠𝐷𝐸𝐹 & ∠𝐷𝐸𝐶 are right angles Definition of right angles Given 𝐷𝐸 ≅ 𝐷𝐸 Reflexive Property ∆𝐷𝐸𝐹≅∆𝐷𝐸𝐶 HL
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Example 5: 𝑃𝑄 ⊥ 𝑆𝑄 Definition of perpendicular
𝑃𝑄 ⊥ 𝑆𝑄 Definition of perpendicular ∆𝑃𝑅𝑆 & ∆𝑃𝑅𝑄 are right triangles Given 𝑃𝑅 ≅ 𝑃𝑅 Reflexive Property ∆𝑃𝑅𝑄 ≅∆𝑃𝑅𝑆 HL
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Try the you try before looking at the answer!
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You Try! 𝑊𝑌 ≅ 𝑋𝑍 𝑊𝑍 ⊥ 𝑍𝑌 𝑋𝑌 ⊥ 𝑍𝑌 Definition of perpendicular
𝑊𝑍 ⊥ 𝑍𝑌 𝑋𝑌 ⊥ 𝑍𝑌 Definition of perpendicular Definition of right angles 𝑍𝑌 ≅ 𝑍𝑌 ∆WYZ ≅ ∆XZY HL
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