Lesson 4.3 and 4.4 Proving Triangles are

Congruentp. 212

Learning Target

I can list the conditions (SAS, SSS) to prove triangles are congruent.

I can identify and use reflexive, symmetric and transitive property in my proof.

How To Find if Triangles are Congruent

Two triangles are congruent if they have:

exactly the same three sides and exactly the same three angles.

But we don't have to know all three sides and all three angles ...usually three out of the six is enough.

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS   (side, side, side)

SSS stands for "side, side, side“ and means that we have two triangles with all three sides equal. For example:

is congruent to:  

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS   (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:is congruent to:  

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA   (angle, side, angle)

ASA stands for "angle, side, angle“ and means that we have two triangles where we know two angles and the included side are equal.

For example:is congruent to:  

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS   (angle, angle, side)

AAS stands for "angle, angle, side“ and means that we have two triangles where we know two angles and the non-included side are equal.

For example:is congruent to:  

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL   (hypotenuse, leg)

HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the other two sides are called "legs")

and  

HL applies only to right angled-triangles!

5. HL   (hypotenuse, leg) It means we have two right-angled

triangles with the same length of hypotenuse and the same length for one of the other

two legs. It doesn't matter which leg since the

triangles could be rotated. For example:

is congruent to If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution ! Don't Use "AAA" !

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

For example: is congruent to 

Without knowing at least one side, we can't be sure if two triangles are congruent..

Goal 2

You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent.

THEOREM

Theorem 4.4 Properties of Congruent Triangles

Reflexive Property of Congruent Triangles

D

E

F

A

B

C

J K

L

Every triangle is congruent to itself.Symmetric Property of Congruent Triangles

Transitive Property of Congruent Triangles

If , then .ABC DEF DEF ABC

If and , then .JKLABC DEF DEF ABC JKL

Proving Triangles are Congruent

1

Using the SAS Congruence Postulate

Prove that AEB DEC.

2

3 AEB DEC SAS Congruence Postulate

21

AE DE, BE CE Given

1 2 Vertical Angles Theorem

Statements

Reasons

D

GA R

Proving Triangles Congruent

MODELING A REAL-LIFE SITUATION

PROVE DRA DRG

SOLUTION

ARCHITECTURE You are designing the window shown in the drawing. Youwant to make DRA congruent to DRG. You design the window so that DR AG and RA RG.

Can you conclude that DRA DRG ?

GIVEN DR AG

RA RG

2

3

4

5

6 SAS Congruence Postulate DRA DRG

1

Proving Triangles Congruent

GivenDR AG

If 2 lines are , then they form 4 right angles.

DRA and DRGare right angles.

Right Angle Congruence Theorem DRA DRG

GivenRA RG

Reflexive Property of CongruenceDR DR

Statements Reasons

D

GA R

GIVEN

PROVE DRA DRG

DR AG

RA RG

Given: SP QR; QP PRProve SPQ SPR

Q P R

S

Statements Reasons1. Given

1. SP QR; QP PR 2. QPS and RPS are right ’s.

2. Def. of

3. QPS PRS 3. Rt. Thm.

4. SP SP 4. Reflexive POC

5. SPQ SPR 5. SAS Post.

Pair-share

Work on classwork on “Congruence Triangle”

Sage and Scribe on #21 to #24