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Learning Target

I can use theorems, postulates or definitions to prove that…

a. vertical angles are congruent.

b. When a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and same-side interior angles are supplementary.

Proving Vertical Angle Theorem

5 and 6 are a linear pair,

GIVEN

6 and 7 are a linear pair

5 7

1

2

3

PROVE

Statements

Reasons

5 and 6 are a linear pair,Given

6 and 7 are a linear pair

5 and 6 are supplementary,Linear Pair Postulate

6 and 7 are supplementary

5 7 Congruent Supplements 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.

IfADandBE,thenCF.

1 2

POSTULATE

POSTULATE 15Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

1

2

3 4

THEOREMS ABOUT PARALLEL LINES

THEOREM 3.4Alternate Interior Angles

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

3

4

m 5 +m 6 = 180°

THEOREMS ABOUT PARALLEL LINES

THEOREM 3.5Consecutive Interior Angles

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

5

6

7 8

THEOREMS ABOUT PARALLEL LINES

THEOREM 3.6Alternate Exterior Angles

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

7

8

jk

THEOREMS ABOUT PARALLEL LINES

THEOREM 3.7Perpendicular Transversal

If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.

Proving the Alternate Interior Angles Theorem

4

3

2

1

p || q

GIVEN

1 2

PROVE

Statements

Reasons

1 3Corresponding Angles Postulate

3 2Vertical Angles Theorem

1 2Transitive property of Congruence

Prove the Alternate Interior Angles Theorem.

SOLUTION

p || q Given

Using Properties of Parallel Lines

m 6 = m 5 = 65°

Vertical Angles Theorem

m 7 =180° – m 5 =115°

Linear Pair Postulate

m 8 = m 5 = 65°

Corresponding Angles Postulate

m 9 = m 7 = 115°

Alternate Exterior Angles Theorem

Given that m 5 = 65°,

find each measure. Tell

which postulate or theorem

you use.

SOLUTION

Using Properties of Parallel Lines

m 4 =125°

Corresponding Angles Postulate

m 4 + (x + 15)° =180°

Linear Pair Postulate

125° + (x + 15)° =180°

Substitute.

x =40°

Subtract.

PROPERTIES OF SPECIAL PAIRS OF ANGLES

Use properties of

parallel lines to findthe value of x.

SOLUTION

Estimating Earth’s Circumference: History Connection

1

m 2

of a circle

50

Over 2000 years ago Eratosthenes estimated Earth’s circumference by using the fact that the Sun’s rays are parallel.

When the Sun shone exactly down a vertical well in Syene, he measured the angle the Sun’s rays made with a vertical stick in Alexandria. He discovered that

Estimating Earth’s Circumference: History Connection

m 1 = m 2

Using properties of parallel lines, he knew that

He reasoned that

1

1

m 2

m 1

of a circle

of a circle

50

50

Estimating Earth’s Circumference: History Connection

575 miles

of a circle

Earth’s circumference

Earth’s circumference

50(575 miles)

Use cross product property

29,000 miles

1

1

m 1

of a circle

50

50

How did Eratosthenes know that m 1 = m 2 ?

The distance from Syene to Alexandria was believed to be 575 miles

Estimating Earth’s Circumference: History Connection

Because the Sun’s rays are parallel,

Angles 1 and 2 are alternate interior angles, so

1 2

By the definition of congruent angles,

How did Eratosthenes know that m 1 = m 2 ?

m 1 = m 2

SOLUTION

Example

Using the Third Angles Theorem

Find the value of x.

SOLUTION

In the diagram, NR and LS.

From the Third Angles Theorem, you know that MT.

So, m M = m T.

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

m M= m T

Third Angles Theorem

60˚ = (2x+ 30)˚

Substitute.

30 = 2x

Subtract 30 from each side.

15 = x

Divide each side by 2.

Goal 2

Learning

Target

Proving Triangles are Congruent

,

,

and

RP

MN

PQ

NQ

QR

QM

PQR

NQM

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

Decide whether the triangles are congruent. Justify your reasoning.

SOLUTION

Paragraph Proof

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

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.

Example

Proving Two Triangles are Congruent

Prove that .

AEB

DEC

A

B

E

|| ,

AB

DC

D

C

E is the midpoint of BC and AD.

GIVEN

.

PROVE

AEB

DEC

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.

,

AB

DC

Example

Proving Two Triangles are Congruent

DEC

Prove that .

AEB

EAB EDC,

A

B

ABE DCE

E

D

C

|| ,

AB

DC

AB

DC

E is the midpoint of AD,

E is the midpoint of BC

,

AE

BE

CE

DE

DEC

AEB

SOLUTION

Given

Alternate Interior Angles Theorem

Vertical Angles Theorem

AEB DEC

Given

Definition of midpoint

Definition of congruent triangles

Goal 2

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 CE Given

1 2Vertical Angles Theorem

3

AEBDEC SAS Congruence Postulate

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

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

Congruent Triangles in a Coordinate Plane

AC FH

ABFG

Use the SSS Congruence Postulate to show that ABCFGH.

SOLUTION

AC = 3 and FH= 3

AB = 5 and FG= 5

Congruent Triangles in a Coordinate Plane

d = (x2 – x1 )2+ (y2 – y1 )2

d = (x2 – x1 )2+ (y2 – y1 )2

BC = (–4 – (–7))2+ (5– 0)2

GH = (6 – 1)2+ (5– 2)2

= 32+ 52

= 52+ 32

= 34

= 34

Use the distance formula to find lengths BC and GH.

Congruent Triangles in a Coordinate Plane

BCGH

BC = 34 and GH= 34

All three pairs of corresponding sides are congruent,

ABCFGH by the SSS Congruence Postulate.

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