Topic 3Congruent Triangles

Unit 2 Topic 4

Information•Two triangles are congruent if they have

the same size and shape. You can turn, flip and/or slide one so it fits exactly on the other. Congruent angles can be marked with symbols, such as an arc, or a dot. Congruent sides can be marked with small line segments, called hatch marks.

Information• If ABC and DEF are congruent, then the

corresponding angles and corresponding sides are equal.

•The symbol for congruence is . Since ABC and DEF are congruent, we write

.

ABC DEF

Information•By definition, if two triangles are

congruent, then all corresponding angles are equal and all corresponding sides are equal. However, to prove two triangles are congruent, you do not need to know that all corresponding sides and all corresponding angles are equal. There are three conditions that can be used to prove two triangles are congruent.

Information Triangle Congruence Conditions

• SSS Congruence Condition▫ If three sides of one triangle are equal to three

sides of another triangle, then the triangles are congruent.

A

B

C P

Q

R

ABC PQR

…..

Information Triangle Congruence Conditions

• SAS Congruence Condition▫ If two sides and the contained angle of one

triangle are equal to two sides and the contained angle of another triangle, then the triangles are congruent

A

B

C P

Q

R

ABC PQR

…..

Information Triangle Congruence Conditions

• ASA Congruence Condition▫ If two angles and the contained side of one

triangle are equal to two angles and the contained side of another triangle, then the triangles are congruent

A

B

C P

Q

R

ABC PQR

…..

Example 1Stating Congruence and Corresponding Angles and Sides.

For each pair of triangles below, state the congruence theorem that proves they are congruent. Then, state the corresponding angles and sides for a).

Try this on your own first!!!!

a) c)

b) d)

A

B C

3.6 cm

3.5 cm

2.1 cm

X

YZ

3.6 cm 2.1 cm

3.5 cm

40˚

50˚

2.2 cm

X

Z Y

A

B C

40˚

50˚

2.2 cm

x

A

B C X

Y Zx

x

A

B C

xX

Y

Z

Example 1a: Solution

a) We can see that all three sides are equal. Therefore, the triangles are congruent by SSS congruence.

A

B C

3.6 cm

3.5 cm

2.1 cm

X

YZ

3.6 cm 2.1 cm

3.5 cm

Corresponding sides: Corresponding

angles:

AB = XY A = X

BC = YZ B = Y

AC = XZ C = Z

Example 1b: Solution

b) Two angles and the contained side of one triangle is equal to two angles and the contained side of another triangle. Therefore, the triangles are congruent by the ASA congruence theorem.Corresponding sides: Corresponding

angles:

AB = XY A = X

BC = YZ B = Y

AC = XZ C = Z

x

A

B C

X

Y Zx

Example 1c: Solution

c) Two sides and the contained angle of one triangle is equal to two sides and the contained angle of another triangle. Therefore, the triangles are congruent by the SAS congruence theorem.Corresponding sides: Corresponding

angles:

AB = XY A = X

BC = YZ B = Y

AC = XZ C = Z

x

A

B C

xX

Y

Z

Example 1d: Solution

d) Since angle B and angle Y are each 90°, we can add to the diagram.

40˚

50˚

2.2 cm

X

Z Y

A

B C

40˚

50˚

2.2 cmTwo angles and the contained side of one triangle is equal to two angles and the contained side of another triangle. Therefore, the triangles are congruent by the ASA congruence theorem.

90°

90°Corresponding sides: Corresponding

angles:

AC = XY A = X

AB = XY B = Y

BC = YZ C = Z

More Information• Using algebra is one way to present a proof. Another way

to present a proof, often used in geometry, is called a two-column proof. A two-column proof is a presentation of a logical argument involving deductive reasoning in which the statements of the argument are in one column and the justification for the statements are written in another column.

• To prove two triangles are congruent you need to provide a logical argument that establishes one of the three congruence conditions: SSS, SAS or ASA. Your proof consists of a set of statements and accompanying reasons. There are many reasons that you can use to justify the statements in your proof.

More Information

You can end a proof with Q.E.D., a Latin phrase that means "which had to be demonstrated”.

Example 2aProving Two Triangles Are Congruent

Try this on your own first!!!!

Given: A = D and AB = DBProve: ∆ABC ∆DBE

Statement Reason

Example 2aProving Two Triangles Are Congruent

Given: A = D and AB = DBProve: ∆ABC ∆DBE

Statement Reason

Given

Given

Opposite angles are equal

ASA Congruency condition

A

S

A

CAB EDB

AB DB

ABC DBE

ABC DBE

Example 2bProving Two Triangles Are Congruent

Try this on your own first!!!!

Statement Reason

Example 2b: SolutionProving Two Triangles Are Congruent

Statement Reason

Given

Given

Common side

SAS Congruency condition

S AS

AB CB

ABD CBD

ABD CBD

BD BD

Example 3a

Statement Reason

Completing a Proof Using DefinitionsGiven: point E is the midpoint AC, point E is the midpoint of BD

Prove: AB = CD

Example 3a: Solution

C

D

A

EB

Statement Reason

BE=DE

CE=AE

By definition of a midpoint

Opposite angles are equal

By definition of a midpoint

SAS Congruency condition

A

S

S

DEC BEA

DEC BEA

Example 3bCompleting a Proof Using Definitions

Given: TP is perpendicular to AC, represented as TP AC

TP bisects ATCProve: AT = CT

Try this on your own first!!!!

T

PA C

Statement Reason

Helpful HintTo bisect is to divide in exactly half.

   

Given: TP is perpendicular to AC, represented as TP AC

TP bisects ATCProve: AT = CT

Example 3b: Solution

T

PA C

Statement Reason

TPA=TPC

TP=TP

ATP=CTP

AT=CT

By definition of a perpendicular

Common side

By definition of a bisect of ATC

ASA Congruency

By congruency

TP is perpendicular to AC meaning they meet at a right angle.

A

S

A

ATP CTP

Example 4Completing a Proof with Parallel Lines

Given: and Prove:

Try this on your own first!!!!

Statement Reason

TU XW TU XWTUV XWV

T U

V

W X

Example 4Completing a Proof with Parallel Lines

Given: and Prove:

Statement Reason

VTU=VXW

TU=XW

VUT=VWX

Alternate interior angles are equal

Given

Alternate interior angles are equal

ASA Congruency

TU XW TU XWTUV XWV

T U

V

W X

TUV XWV

A

S

A

Example 5Completing a Proof Using Supplementary Angles

Given: BC=ED, OBA= OEF, and OCB= ODE. Prove: BOC = EOD.

Try this on your own first!!!!

Statement ReasonF

O E

D

CB

A

Example 5: SolutionCompleting a Proof Using Supplementary Angles

Given: BC=ED, OBA= OEF, and OCB= ODE. Prove: BOC = EOD.

Statement Reason

ABO=FEO

OBC=OED

BC=ED

BCO=EDO

BOC=EOD

Given

Supplementary of the given angles

Given

Given

ASA Congruency

By Congruency

F

O E

D

CB

A

BCO EDO A

S

A

Need to Know:• Congruent triangles have the same size and shape.

• To prove that triangles are congruent, it must be shown that the corresponding sides and corresponding angles in the triangles are equal.

• To do this, the following Triangle Congruence Conditions are used:

▫SSS Congruence Theorem▫SAS Congruence Theorem▫ASA Congruence Theorem

You’re ready! Try the homework from this section.