Concepts, Theorems and Postulates that can be use to prove that triangles

are congruent.

Learning Target

• I can identify and use reflexive, symmetric and transitive property in proving two triangles are congruent.

• I can use theorems about line and angles in my proof.

Corresponding Anglesand

Corresponding Sides

Goal 1

Identifying Congruent Figures

Two geometric figures are congruent if they have exactly the same size and shape.

Each of the red figures is congruent to the other red figures.

None of the blue figures is congruent to another blue figure.

Learning Target

Goal 1

Identifying Congruent Figures

When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent.

Corresponding Angles Corresponding Sides

A P B Q C R

BC QR

RPCA

AB PQ

For the triangles below, you can write , which reads “triangle ABC is congruent to triangle PQR.” The notation shows the congruence and the correspondence.

ABC PQR

There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write .BCA QRP

Learning Target

Example

Naming Congruent Parts

The two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts.

SOLUTION

Angles:

Sides:

D R, E S, F T

, , RS DE TRFD ST EF

The diagram indicates that .

The congruent angles and sides are as follows.

DEF RST

Example

Using Properties of Congruent Figures

In the diagram, NPLM EFGH.

Find the value of x.

SOLUTION

You know that .GHLM

So, LM = GH.

8 = 2 x – 3

11 = 2 x

5.5 = x

Example

Using Properties of Congruent Figures

In the diagram, NPLM EFGH.

Find the value of x.

SOLUTION

You know that .GHLM

So, LM = GH.

8 = 2 x – 3

11 = 2 x

5.5 = x

Find the value of y.

You know that N E.

So, m N = m E.

72˚ = (7y + 9)˚

63 = 7y

9 = y

SOLUTION

Theorems and Postulates on Congruent Angles

(Transitive, Reflexive and Symmetry Theorem of Angle Congruence)

CONGRUENCE OF ANGLES

THEOREM

THEOREM 2.2 Properties of Angle Congruence

Angle congruence is r ef lex ive, sy mme tric, and transitive.Here are some examples.

TRANSITIVE If A B and B C, then A C

SYMMETRIC If A B, then B A

REFLEX IVE For any angle A, A A

Transitive Property of Angle Congruence

Prove the Transitive Property of Congruence for angles.

SOLUTION To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C.

GIVEN A B, PROVE A C

A

B

C

B C

Transitive Property of Angle Congruence

GIVEN A B,

B C

PROVE A C

Statements Reasons

1

2

3

4

m A = m B Definition of congruent angles

5 A C Definition of congruent angles

A B, Given

B C

m B = m C Definition of congruent angles

m A = m C Transitive property of equality

Using the Transitive Property

This two-column proof uses the Transitive Property.

Statements Reasons

2

3

4

m 1 = m 3 Definition of congruent angles

GIVEN m 3 = 40°, 1 2, 2 3

PROVE m 1 = 40°

1

m 1 = 40° Substitution property of equality

1 3 Transitive property of Congruence

Givenm 3 = 40°, 1 2,

2 3

Right Angle Theorem

Proving Right Angle Congruence Theorem

THEOREM

Right Angle Congruence Theorem

All right angles are congruent.

You can prove Right Angle CongruenceTheorem as shown.

GIVEN 1 and 2 are right angles

PROVE 1 2

Proving Right Angle Congruence Theorem

Statements Reasons

1

2

3

4

m 1 = 90°, m 2 = 90° Definition of right angles

m 1 = m 2 Transitive property of equality

1 2 Definition of congruent angles

GIVEN 1 and 2 are right angles

PROVE 1 2

1 and 2 are right angles Given

Congruent Supplements Theorem

(Supplementary Angles)

PROPERTIES OF SPECIAL PAIRS OF ANGLES

THEOREMS

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.

1 2

3

PROPERTIES OF SPECIAL PAIRS OF ANGLES

THEOREMS

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.

1 233

If m 1 + m 2 = 180°

m 2 + m 3 = 180°

and

1

then

1 3

Congruent Complements Theorem

(Complementary Angles)

PROPERTIES OF SPECIAL PAIRS OF ANGLES

THEOREMS

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent.

45

6

PROPERTIES OF SPECIAL PAIRS OF ANGLES

THEOREMS

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent.

4

If m 4 + m 5 = 90°

m 5 + m 6 = 90°

and

then

4 6

566

4

Proving Congruent Supplements Theorem

Statements Reasons

1

2

GIVEN 1 and 2 are supplements

PROVE 2 3

3 and 4 are supplements

1 4

1 and 2 are supplements Given

3 and 4 are supplements

1 4

m 1 + m 2 = 180° Definition of supplementary anglesm 3 + m 4 = 180°

Proving Congruent Supplements Theorem

Statements Reasons

3

GIVEN 1 and 2 are supplements

PROVE 2 3

3 and 4 are supplements

1 4

4

5 m 1 + m 2 = Substitution property of equalitym 3 + m 1

m 1 + m 2 = Transitive property of equalitym 3 + m 4

m 1 = m 4 Definition of congruent angles

Proving Congruent Supplements Theorem

Statements Reasons

GIVEN 1 and 2 are supplements

PROVE 2 3

3 and 4 are supplements

1 4

6

7

m 2 = m 3 Subtraction property of equality

2 3 Definition of congruent angles

Linear Pair Postulate

POSTULATE

Linear Pair Postulate

If two angles for m a linear pair, then they are supplementary.

m 1 + m 2 = 180°

PROPERTIES OF SPECIAL PAIRS OF ANGLES

Proving Vertical Angle Theorem

THEOREM

Vertical Angles Theorem

Vertical angles are congruent

1 3, 2 4

Proving Vertical Angle Theorem

PROVE 5 7

GIVEN 5 and 6 are a linear pair,

6 and 7 are a linear pair

1

2

3

Statements Reasons

5 and 6 are a linear pair, Given6 and 7 are a linear pair

5 and 6 are supplementary, Linear Pair Postulate6 and 7 are supplementary

5 7 Congruent Supplements Theorem

Third Angles Theorem

Goal 1

The Third Angles Theorem below follows from the Triangle Sum Theorem.

THEOREM

Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

If A D and B E, then C F.

Example

Using the Third Angles Theorem

Find the value of x.

SOLUTION

In the diagram, N R and L S.

From the Third Angles Theorem, you know that M T. So, m M = m T.

From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚.

m M = m T

60˚ = (2 x + 30)˚

30 = 2 x

15 = x

Third Angles Theorem

Substitute.

Subtract 30 from each side.

Divide each side by 2.

Goal 2

SOLUTION

Paragraph Proof

From the diagram, you are given that all three corresponding sides are congruent.

, NQPQ ,MNRP QMQR and

Because P and N have the same measures, P N.

By the Vertical Angles Theorem, you know that PQR NQM.

By the Third Angles Theorem, R M.

Decide whether the triangles are congruent. Justify your reasoning.

So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, .PQR NQM

Proving Triangles are CongruentLearning Target

Example

Proving Two Triangles are Congruent

A B

C D

E

|| , DCAB ,

DCAB E is the midpoint of BC and AD.

Plan for Proof Use the fact that AEB and DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent.

GIVEN

PROVE .AEB DEC

Prove that .AEB DEC

Example

Proving Two Triangles are Congruent

Statements Reasons

EAB EDC, ABE DCE

AEB DEC

E is the midpoint of AD,E is the midpoint of BC

,DEAE CEBE

Given

Alternate Interior Angles Theorem

Vertical Angles Theorem

Given

Definition of congruent triangles

Definition of midpoint

|| ,DCAB DCAB

SOLUTION

AEB DEC

A B

C D

E

Prove that .AEB DEC

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

SSS AND SAS CONGRUENCE POSTULATES

If all six pairs of corresponding parts (sides and angles) arecongruent, then the triangles are congruent.

and thenIfSides are congruent

1. AB DE

2. BC EF

3. AC DF

Angles are congruent

4. A D

5. B E

6. C F

Triangles are congruent

ABC DEF

SSS AND SAS CONGRUENCE POSTULATES

POSTULATE

POSTULATE: Side - Side - Side (SSS) Congruence Postulate

Side MN QR

Side PM SQ

Side NP RS

If

If three sides of one triangle are congruent to three sidesof a second triangle, then the two triangles are congruent.

then MNP QRS

S

S

S

Using the SSS Congruence Postulate

Prove that PQW TSW.

Paragraph Proof

SOLUTION

So by the SSS Congruence Postulate, you

know that

PQW TSW.

The marks on the diagram show that PQ

TS,

PW TW, and QW SW.

POSTULATE

SSS AND SAS CONGRUENCE POSTULATES

POSTULATE: Side-Angle-Side (SAS) Congruence Postulate

Side PQ WX

Side QS XY

then PQS WXYAngle Q X

If

If two sides and the included angle of one triangle arecongruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

A

S

S

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

Congruent Triangles in a Coordinate Plane

AC FH

AB FGAB = 5 and FG = 5

SOLUTION

Use the SSS Congruence Postulate to show that ABC

FGH.

AC = 3 and FH = 3

Congruent Triangles in a Coordinate Plane

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= 3 2 + 5

2

= 34

BC = (– 4 – (– 7)) 2 + (5 – 0 )

2

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= 5 2 + 3

2

= 34

GH = (6 – 1) 2 + (5 – 2 )

2

Use the distance formula to find lengths BC and GH.

Congruent Triangles in a Coordinate Plane

BC GH

All three pairs of corresponding sides are congruent,

ABC FGH by the SSS Congruence Postulate.

BC = 34 and GH = 34