3.8 HL

Objectives: • Use HL in proofs to prove triangles

congruent.

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. (HL)

Hypotenuse-Leg Congruence Theorem (HL)

A

B

C

If and , then .AB RS BC ST ABC RST

R

S

T

1.right triangle

2.hypotenuse

3.leg

5 ways to prove triangles congruent:

1. SSS 2. SAS 3. ASA 4. AAS 5. HL (only rt. ∆’s)

Example 1:

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

Given

Given

Definition of perpendicularDCE and ACB are right angles

is the midpoint of C BD

Definition of midpoint

Definition of congruent segmentsDC CB

B

A

CD

E

Given : , ,

is the midpoint of

Prove :

AE BD AB ED

C BD

ABC EDC

AE BD

DC = CB

AB ED Given

HL∆ABC ∆EDC

∆DCE and ∆ACB are right triangles Definition of right triangle

Example 2:

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

All right angles are congruent

Vertical Angles Theorem

Given

Definition of altitudeBGF, ECF, BGA, ECD are right angles

Definition of midpoint

Definition of congruent segmentsBF EF

BGF ECF

BF = EF

Given : and are altitudes,

is the midpoint of ,

Prove :

BG EC

F BE AB ED

BAG EDC

B

E

C D

A G F

Continued on next slide

and are altitudesBG EC

BFG EFC

is the midpoint of F BE Given

Example 2:

Statements Reasons

8. 8.

9. 9.

10. 10.

11. 11.

12. 12.HL

CPCTC

GivenAB ED

Given : and are altitudes,

is the midpoint of ,

Prove :

BG EC

F BE AB ED

BAG EDC

B

E

C D

A G F

BG EC

∆BAG ∆EDF

AAS∆BGF ∆ECF

Definition of right triangle∆BAG and ∆EDF are right triangles