4-6 Congruence in Right Triangles

Quick Review

If the 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 by the Hypotenuse-Leg (HL) Theorem.

Example

Which two triangles are congruent? Explain.

Since Δ A B C cap delta eh b c  and Δ X Y Z cap delta x y z  are right triangles with congruent legs, and B C ¯ ≅ Y Z ¯ , Δ A B C ≅ Δ X Y Z b c bar , approximately equal to . y z bar , comma . cap delta eh b c approximately equal to cap delta x y z  by HL.

Exercises

Write a proof for each of the following.

29. Given: L N ¯ ⊥ K M ¯ , K L ¯ ≅ M L ¯ l n bar , up tack . k m bar , comma . k l bar , approximately equal to . m l bar

    Prove: Δ K L N ≅ Δ M L N cap delta k l n approximately equal to cap delta m l n

    

30. Given: P S ¯ ⊥ S Q ¯ , R Q ¯ ⊥ Q S ¯ , P Q ¯ ≅ R S ¯ p s bar , up tack . s q bar , comma . r q bar , up tack . q s bar , comma . p q bar , approximately equal to . r s bar

    Prove: Δ P S Q ≅ Δ R Q S cap delta p s q approximately equal to cap delta r q s

    

4-7 Congruence in Overlapping Triangles

Quick Review

To prove overlapping triangles congruent, you look for the common or shared sides and angles.

Example

Separate and redraw the overlapping triangles. Label the vertices.

Exercises

Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL.

Table of Contents