SECTION 3.8- THE HL POSTULATE

By: Eric OnofreyTyler JulianDrew Kuzma

Why Do You Need To Know This?

Let’s say you need to prove triangles congruent

But there is not enough information to use SAS, ASA, or SSS.

Now you’re stuck right?.....WRONG! The Hypotenuse Leg Postulate is another

method of proving triangles congruent

What is the HL Postulate?

HL Postulate: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent.

Or for short, (HL)

How and When to Use It

The HL Postulate only works with right triangles.

When used in a proof, you must establish the two are right triangles.

So after you do that, you get the legs and hypotenuses congruent and you’re done!

Sample ProblemA

B C

D

E F

Given: AB ┴ BC

DE ┴ EF

AB DE

AC DF

Statements

1. AB ┴ BC

2. DE ┴ EF

3. AB DE

4. AC DF

5. <ABC, <DEF are right

<s

6. Triangle ABC, triangle

DEF are right

triangles

7. Triangle ABC

triangle DEF

Reasons

1. Given

2. Given

3. Given

4. Given

5. ┴ Lines form right <s

6. If a triangle has one

right <, then it is a

right triangle

7. HL ( 3, 4, 6)

Prove: Triangle ABC triangle DEF

Practice Problem #1 Statements1.F is the midpoint of AD 2. 3. 4.

5. <EFA, < EFD are rt <s6. Triangle EFD and triangle

EFA are right triangles7. Triangle EFD is congruent

to triangle EFA8. <AEF is congruent to <

DEF

Reasons1. Given 2. Given3. Given4. If a pt if a midpoint of a seg,

then it divides the seg into 2 congruent segs.

5. Perpendicular lines form rt <s6. If a triangle has one right <,

then it is right 7. HL (2, 4, 6)

8. CPCTC

E

A F D

B C

Given: F is the midpoint of AD

Prove: <AEF congruent to < DEF

EDEAADEF

EDEAADEF FDAF

Practice Problem #2

Statements1. ABCD is a rectangle2. AC is congruent to BD

3. AB is congruent to DC

4. <ABC, <DCB are right5. Triangle ABC, Triangle

DCB are right triangles6.Triangle ABC is

congruent to triangle DCB

7. <EBC is congruent to < ECB

8. Triangle BEC is an isosceles triangle

Reasons1. Given2. Rectangle implies diagonals

congruent3. Rectangle implies opposite

sides congruent4. Rectangle implies right angles

5. If a triangle has one right angle, then it is right.

6. HL (2,3 5,)

7. CPCTC

8. If two <s are congruent then the triangle is isosceles.

A D E

B C

Given: ABCD is a rectangle

Prove: Triangle BEC is an isosceles triangle

Practice Problem #3

Statements1. ABDE is a rectangle

2.

3.

4.<ABC, < EDC are right <s

5.Triangle ABC, triangle EDC are right triangles

6.Triangle ABC is congruent to triangle EDC

7.<BAC is congruent to <DEC8. <BAE, < DEA are right <s9. <BAE is congruent to <DEA

10. <CAE is congruent to <CEA

Reasons

1. Given2. Given

3. Rectangle implies opposite sides congruent

4. Rectangle implies right <s5. If a triangle has one right

< then it is a right triangle

6. HL (2,3, 5)

7. CPCTC8. Rectangle implies right <s9. Right angles are

congruent10. Subtraction

A E

B C D Given: ABDE is a

rectangle

ECAC

Prove: <AEC is congruent to < EAC

ECAC

EDAB

Practice Problem #4

Statements1. ABCD is a square2. BD Bisects AC

3.

4.5.

6. <BEC, < AED are right <s

7. Triangle BEC and triangle AED are right triangles

8. Triangle BEC is congruent to triangle AED

Statements1. Given2. Square implies diagonals

bisect 3. If a seg is bisected, then it is

divided into 2 congruent segs

4. Square implies sides congruent

5. Square implies diagonals perpendicular

6. Perpendicular lines form right <s

7. If a triangle has one right < then it is a right triangle

8. HL (3, 4, 7)

A

E

B D

C Given: ABCD is a square

Prove: Triangle AED is congruent to triangle BEC

ECAE

BDAC

ADBC

Practice Problem #5

Statements1.Circle D2. BD is an altitude of

Triangle ABC3. 4. <ADB, <CDB are right <s

5. Triangle ADB and triangle CDB are right triangles

6. 7. Triangle ADB is congruent

to triangle CDB

Reasons1. Given2. Given

3. Given4. If a seg is an altitude,

then it is drawn from a triangle vertex and forms right <s with the opposite side.

5. If a triangle has one right <, then it is a right triangle

6. All radii of a circle are congruent.

7. HL (3, 5, 6)

A D C

BGiven: Circle D

BD is an altitude of triangle ABC

Prove: Triangle ABD is congruent to triangle CBD

BCAB

BCAB

DCAD

Practice Problem #6

Statements1. Circle A2. 3. 4. <ACB, <ACD are right <s

5, Triangle ABC, triangle ADC are right triangles

6. 7. Triangle ABC is congruent

to triangle ADC

Reasons1. Given2. Given3. Given4. Perpendicular lines

form right <s5. If a triangle has one

right <, then it is a right triangle

6. All radii of a circle are congruent

7. HL( 2, 5, 6)

B C D

A

Given: Circle A

Prove: Triangle ABC is congruent to triangle ADC

BDAC

BDAC CDBC

CDBC

ADAB

Works Cited

CliffsNotes.com. Congruent Triangles. 18 Jan 2011<http://www.cliffsnotes.com/study_guide/topicArticleId-18851,articleId-18788.html>.

"Geometry: Congruent Triangles - CliffsNotes." Get Homework Help with CliffsNotes Study Guides - CliffsNotes. Web. 18 Jan. 2011. <http://www.cliffsnotes.com/study_guide/Congruent-Triangles.topicArticleId-18851,articleId-18788.html>.