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Lesson 4.3 and 4.4 Proving Triangles are Congruent

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Lesson 4.3 and 4.4 Proving Triangles are Congruent

p. 212

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.

- Two triangles are congruent if they have:
- exactly the same three sides and
- exactly the same three angles.

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

SSS stands for "side, side, side“

and means that we have two triangles

with all three sides 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.

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

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

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:

HL applies only to right angled-triangles!

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

and

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.

- It means we have two right-angled triangles with
- the same length of hypotenuseand
- thesame 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

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

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

Proving Triangles are Congruent

E

B

D

A

F

C

DEF

DEF

ABC

DEF

ABC

ABC

DEF

JKL

ABC

JKL

If , then .

L

If and , then .

J

K

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 4.4Properties of Congruent Triangles

Reflexive Property of Congruent Triangles

Every triangle is congruent to itself.

Symmetric Property of Congruent Triangles

Transitive Property of Congruent Triangles

Using the SAS Congruence Postulate

Prove that

AEBDEC.

1

2

1

2

Statements

Reasons

AE DE, BE CEGiven

1 2Vertical Angles Theorem

3

AEBDEC SAS Congruence Postulate

Proving Triangles Congruent

ARCHITECTURE You are designing the window shown in the drawing. You

want to make DRAcongruent to DRG. You design the window so that

DRAG and RARG.

D

A

G

R

GIVEN

DRAG

RARG

DRADRG

PROVE

MODELING A REAL-LIFE SITUATION

Can you conclude that DRADRG?

SOLUTION

Proving Triangles Congruent

GIVEN

RARG

DRADRG

PROVE

1

2

6

3

4

5

Statements

Reasons

Given

DRAG

If 2 lines are, then they form

4 right angles.

DRA and DRG

are right angles.

Right Angle Congruence Theorem

DRADRG

DRAG

Given

RARG

DRDR

Reflexive Property of Congruence

SAS Congruence Postulate

DRADRG

D

A

R

G

S

StatementsReasons

1. 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.

Q

R

P

Work on classwork on “Congruence Triangle”

Sage and Scribe on #21 to #24