Learning Target #17 I can use theorems, postulates or definitions to prove that… a. vertical angles are congruent. b. When a transversal crosses parallel.

Learning Target #17

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

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


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.

PROPERTIES OF PARALLEL LINES

POSTULATE

POSTULATE 15 Corresponding Angles Postulate

1

2

1 2

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


THEOREMS ABOUT PARALLEL LINES

THEOREM 3.4 Alternate Interior Angles

3

4

3 4

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



THEOREM 3.5 Consecutive Interior Angles

5

6

m 5 + m 6 = 180°

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



THEOREM 3.6 Alternate Exterior Angles

7

8

7 8

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



THEOREM 3.7 Perpendicular Transversal

j k

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

Proving the Alternate Interior Angles Theorem

Prove the Alternate Interior Angles Theorem.

SOLUTION

GIVEN p || q

p || q Given

Statements Reasons

1

2

3

4

PROVE 1 2

1 3 Corresponding Angles Postulate

3 2 Vertical Angles Theorem

1 2 Transitive property of Congruence

Using Properties of Parallel Lines

SOLUTION

Given that m 5 = 65°, find each measure. Tellwhich postulate or theoremyou use.

Linear Pair Postulatem 7 = 180° – m 5 = 115°

Alternate Exterior Angles Theoremm 9 = m 7 = 115°

Corresponding Angles Postulatem 8 = m 5 = 65°

m 6 = m 5 = 65° Vertical Angles Theorem

Using Properties of Parallel Lines

Use properties ofparallel lines to findthe value of x.

SOLUTION

Corresponding Angles Postulatem 4 = 125°

Linear Pair Postulatem 4 + (x + 15)° = 180°

Substitute.125° + (x + 15)° = 180°

PROPERTIES OF SPECIAL PAIRS OF ANGLES

Subtract.x = 40°

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 2 150 of a circle


m 2 150 of a circle

Using properties of parallel lines, he knew that

m 1 = m 2

He reasoned that

m 1 150 of a circle

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


m 1 150 of a circle

Earth’s circumference

150 of a circle

575 miles

Earth’s circumference 50(575 miles)

Use cross product property

29,000 miles

How did Eratosthenes know that m 1 = m 2 ?


How did Eratosthenes know that m 1 = m 2 ?

SOLUTION

Angles 1 and 2 are alternate interior angles, so

1 2

By the definition of congruent angles,

m 1 = m 2

Because the Sun’s rays are parallel,

Example

Using the Third Angles Theorem

Find the value of x.

SOLUTIO

NIn 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


Substitute.

Subtract 30 from each side.

Divide each side by 2.

Goal 2

SOLUTIO

NParagraph ProofFrom 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 CongruentLearningTarget

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

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


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.


BC GH

All three pairs of corresponding sides are congruent, ABC FGH by the SSS Congruence Postulate.

BC = 34 and GH = 34

Learning Target #17 I can use theorems, postulates or definitions to prove that… a. vertical angles are congruent. b. When a transversal crosses parallel.

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