4.9 (M1) Prove 4.9 (M1) Prove Triangles Congruent Triangles Congruent

by SAS & HLby SAS & HL

VocabularyVocabulary

In a right triangle, the sides adjacent to In a right triangle, the sides adjacent to the right angle are the the right angle are the legs.legs.

The side opposite the right angle is the The side opposite the right angle is the hypotenusehypotenuse..

Side-Angle-Side (SAS) Congruence Side-Angle-Side (SAS) Congruence Postulate: Postulate: If two sides and the If two sides and the included angle of one triangle are included angle of one triangle are congruent to two sides and the congruent to two sides and the included angle of another triangle, the included angle of another triangle, the two triangles are congruent.two triangles are congruent.

Hypotenuse-Leg (HL) Hypotenuse-Leg (HL) Congruence Theorem – Congruence Theorem – If the If the hypotenuse and one leg of a right hypotenuse and one leg of a right triangle are congruent to the triangle are congruent to the hypotenuse and leg of another right hypotenuse and leg of another right triangle, the triangles are congruent.triangle, the triangles are congruent.

ANSWER Yes; HL Thm.

Tell whether the pair of triangles is congruent or not and why.

Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.

1. ABE, CBD

ANSWER SAS Post.

Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.

2. FGH, HJK

ANSWER HL Thm.

State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate.

3. ST YZ, RS XY

ANSWER S Y.

EXAMPLE 1 Use the SAS Congruence Postulate

Write a proof.

GIVEN

PROVE

STATEMENTS REASONS

BC DA, BC AD

ABC CDA

1. Given1. BC DAS

Given2. 2. BC AD

3. BCA DAC 3. Alternate Interior Angles Theorem

A

4. 4. AC CA Reflexive Property of Congruence

S

ABC CDA 5. SAS

EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem

Write a proof.

SOLUTION

Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram.

GIVEN WY XZ, WZ ZY, XY ZY

PROVE WYZ XZY

STATEMENTS REASONS

EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem

1. WY XZ 1. Given

4. 4. Definition of a right triangle

WYZ and XZY are right triangles.

L ZY YZ5. 5. Reflexive Property of Congruence

6. WYZ XZY 6. HL Congruence Theorem

3. 3. Definition of linesZ and Y are right angles

2. 2. WZ ZY, XY ZY Given