1. Congruence in Right Triangles
Skill 24

2. 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.

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. #24: Congruence in Right Triangles
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