Congruency and TrianglesFeaturing Right TrianglesSection 4.6

Lesson ObjectivesLearn the properties of congruent right triangles.Learn the properties of perpendicular bisectors.

Review: Right TrianglesRecall that right triangles are composed of a hypotenuse which is opposite the right angle and two legs. If the two legs are not equal in length they are referred to as the short leg and long leg of the triangle.

Theorem 4-5: The Perpendicular BisectorTheorem:If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base.If ≅ and ∠YXB ≅ ∠ZXB…Then ⊥ and ≅

Theorem 4-6: The Hypotenuse-Leg (HL) TheoremTheorem:If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.If ΔABC and ΔDEF are right triangles and ≅ and ≅ …Then ΔABC ≅ ΔDEF

Theorem 4-6: The Hypotenuse-Leg (HL) Theorem

Theorem 4-6: The Hypotenuse-Leg (HL) TheoremConditions: There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs

Theorem 4-6: The Hypotenuse-Leg (HL) TheoremExample: Given: ∠YBX and ∠ZBX are right angles. by the reflexive property of congruence. ΔXBY ΔXBZ by the HL Theorem

Theorem 4-6: The Hypotenuse-Leg (HL) TheoremTry it! Given: ∠PRS and ∠RPQ are right angles. by the reflexive property of congruence. ΔPRS ΔRPQ by the HL Theorem

PracticeWhat additional information do you need to prove the triangles are congruent by HL? ΔABC ≅ ΔXYZ

Conditions: There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs

PracticeWhat additional information do you need to prove the triangles are congruent by HL? ΔABC ≅ ΔDCB Conditions:

There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs

PracticeWhat additional information do you need to prove the triangles are congruent by HL? ΔSTR ≅ ΔPQN Conditions: There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs

PracticeFor what values of x and y are the triangles congruent by HL?

Conditions: There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs

PracticeFor what values of x and y are the triangles congruent by HL?

Conditions: There are two right triangles The triangles have congruent hypotenuses There is one pair of congruent legs

Proofs using the HL Theorem

Exit SlipSelf-assessment: at the bottom of your answer sheet write down how well you understand this concept from 1 : 41 (don’t understand) 2 (kind of understand) 3 (understand but may need a little help)4 (understand and can explain it to others)