Congruence in Right Triangles Skill 24

Objective HSG-SRT.5: Students are responsible for proving right triangles are congruent using the Hypotenuse-Leg Theorem and find and use relationships in similar right triangles.

Theorem 22: Hypotenuse Leg Theorem HL Theorem If hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. P Q X R Y Z If ∆𝑷𝑸𝑹 and ∆𝑿𝒀𝒁 are right triangles 𝑷𝑹 ≌ 𝑿𝒁 , and 𝑷𝑸 ≌ 𝑿𝒀 Then ∆𝑨𝑩𝑪≌∆𝑫𝑬𝑭

Conditions for HL Theorem There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs P Q X R Y Z

Example 1; Using the HL Theorem Given: ∠𝐴𝐷𝐶 and ∠𝐵𝐷𝐶 are rt. ∠’s in a rt. ∆ and 𝐴𝐶 ≅ 𝐵𝐶 Prove: ∆𝐴𝐷𝐶≅∆𝐵𝐷𝐶 Statement Reason 1) ∠𝐴𝐷𝐶 & ∠𝐵𝐷𝐶 are right angles in a right triangle and 𝐴𝐶 ≅ 𝐵𝐶 1) Given 2) 𝐷𝐶 ≌ 𝐷𝐶 2) Reflexive Prop. A B C D 3) ∆𝐴𝐷𝐶≅∆𝐵𝐷𝐶 3) HL Theorem

Example 2; Using the HL Theorem Given: ∠𝑃𝑅𝑆 & ∠𝑅𝑃𝑄 are rt. ∠’s in a right ∆ and 𝑆𝑃 ≅ 𝑄𝑅 Prove: ∆𝑃𝑅𝑆≅∆𝑅𝑃𝑄 Statement Reason 1) ∠𝑃𝑅𝑆 & ∠𝑅𝑃𝑄 are right angles in a right triangle and 𝑆𝑃 ≅ 𝑄𝑅 1) Given 2) 𝑃𝑅 ≌ 𝑃𝑅 2) Reflexive Prop. 3) ∆𝑃𝑅𝑆≅∆𝑅𝑃𝑄 3) HL Theorem S R P Q

Example 3; Writing a Proof with HL Theorem Given: 𝐵𝐸 bisects 𝐴𝐷 at C, 𝐴𝐵 ⏊ 𝐵𝐶 , 𝐷𝐸 ⏊ 𝐸𝐶 , 𝐴𝐵 ≅ 𝐷𝐸 Prove: ∆𝐴𝐵𝐶≅∆𝐷𝐸𝐶 Statement Reason 1) 𝐵𝐸 bisects 𝐴𝐷 at C, 𝐴𝐵 ⏊ 𝐵𝐶 , 𝐷𝐸 ⏊ 𝐸𝐶 , 𝐴𝐵 ≅ 𝐷𝐸 1) Given 2) ∠𝐵 𝑎𝑛𝑑 ∠𝐸 are rt. ∠’s 2) Def. Perpendicular 3) 𝐴𝐶 ≅ 𝐷𝐸 3) Def. Bisector 4) ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐶 are right triangles B C A D E 4) Def. Right ∆ 5) ∆𝐴𝐵𝐶≅∆𝐷𝐸𝐶 5) HL Theorem

Example 4; Writing a proof with HL Theorem Given: 𝐶𝐷 ≅ 𝐸𝐴 , 𝐴𝐷 is the perp.bisector of 𝐶𝐸 Prove: ∆𝐶𝐵𝐷≅∆𝐸𝐵𝐴 Statement Reason 1) 𝐶𝐷 ≅ 𝐸𝐴 , 𝐴𝐷 is the perpendicular bisector of 𝐶𝐸 1) Given 2) 𝐴𝐵 ≌ 𝐷𝐵 and ∠𝑃𝑅𝑆 & ∠𝑅𝑃𝑄 are right angles 2) Def. Perp. Bisector 3) ∆𝐶𝐵𝐷 and ∆𝐸𝐵𝐴 are right triangles 3) Def. Right ∆ E B C A D 4) ∆𝐶𝐵𝐷≅∆𝐸𝐵𝐴 4) HL Theorem

#24: Congruence in Right Triangles Questions? Summarize Notes Homework Video Quiz