geometry angles formed by parallel lines and transversals

CONFIDENTIAL 1

GeometryGeometry

Angles formed by Angles formed by Parallel Lines and Parallel Lines and

TransversalsTransversals

CONFIDENTIAL 2

Warm UpWarm Up

Give an example of each angle pair.

1) Alternate interior angles 2) Alternate exterior angles

3)Same side interior angles

CONFIDENTIAL 3

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed.

1 23 4

5 67 8

There are several special pairs of angles formed from this figure.

Vertical pairs: Angles 1 and 4  Angles 2 and 3  Angles 5 and 8  Angles 6 and 7

CONFIDENTIAL 4

Supplementary pairs:

Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7

1 23 4

5 67 8

Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs.

There are three other special pairs of angles. These pairs are congruent pairs.

CONFIDENTIAL 5

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

Corresponding angle postulateCorresponding angle postulate

1 2 3 45 6 7 8

p q

t

1 3 2 4 5 7 6 8

CONFIDENTIAL 6

Using the Corresponding angle postulateUsing the Corresponding angle postulate

Find each angle measure.

800 x0

B

C

A

A) m( ABC)

x = 80 corresponding angles

m( ABC) = 800

CONFIDENTIAL 7

B) m( DEF)

(2x-45)0 = (x+30)0 corresponding angles

m( DEF) = (x+30)0

(2x-45)0

EF

D

(x+30)0

x – 45 = 30 subtract x from both sides

x = 75 add 45 to both sides

= (75+30)0

= 1050

CONFIDENTIAL 8

Now you try!

1) m( DEF)

RSx0

1180

Q

CONFIDENTIAL 9

Remember that postulates are statements that are accepted without proof. Since the

Corresponding Angles postulate is given as a postulate, it can be used to prove the next

three theorems.

CONFIDENTIAL 10

Alternate interior angles theoremAlternate interior angles theorem

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

1 3 2 4

1 2

4 3

Theorem

Hypothesis Conclusion

CONFIDENTIAL 11

Alternate exterior angles theoremAlternate exterior angles theorem

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

5 7 6 8

5 6

8 7

Theorem

Hypothesis Conclusion

CONFIDENTIAL 12

Same-side interior angles theoremSame-side interior angles theorem

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

m 1 + m 4 =1800 m 2 + m 3 =1800

Theorem

Hypothesis Conclusion

1 2

4 3

CONFIDENTIAL 13

Alternate interior angles theoremAlternate interior angles theorem

1 2

3m

lGiven: l || m

Prove: 2 3

Proof:

1 3l || m

Given Corresponding angles

2 3

2 1

Vertically opposite angles

CONFIDENTIAL 14

A) m( EDF)

Finding Angle measuresFinding Angle measures

1250

B

C

A

x0

D

E F

m( DEF) = 1250

x = 1250

Alternate exterior angles theorem

Find each angle measure.

CONFIDENTIAL 15

B) m( TUS)

13x0 + 23x0 = 1800 Same-side interior angles theorem

m( TUS) = 23(5)0

36x = 180 Combine like terms

x = 5 divide both sides by 36

= 1150

13x0 23x0

U

T

S

R

Substitute 5 for x

CONFIDENTIAL 16

2) Find each angle measure.

Now you try!

B C

ED

(2x+10)0A

(3x-5)0

CONFIDENTIAL 17

A treble string of grand piano are parallel. Viewed from above, the bass strings form transversals to the treble

string. Find x and y in the diagram.

(25x+5y)0

(25x+4y)0

1200

1250

By the Alternative Exterior Angles Theorem, (25x+5y)0 = 1250

By the Corresponding Angles Postulates, (25x+4y)0 = 1200

(25x+5y)0 = 1250

- (25x+4y)0 = 1200

y = 5

25x+5(5) = 125

x = 4, y = 5

Subtract the second equation from the first equation

Substitute 5 for y in 25x +5y = 125. Simplify and solve for x.

CONFIDENTIAL 18

3) Find the measure of the acute angles in the diagram.

Now you try!

(25x+5y)0

(25x+4y)0

1200

1250

CONFIDENTIAL 19

Assessment

Find each angle measure:

1270

x0

KJ

L

2) m( BEF)

(7x-14)0

(4x+19)0

G

AABC

FD

H

E

1) m( JKL)

CONFIDENTIAL 20

Find each angle measure:

1

3) m( 1)

(3x+9)0

6x0

A

B

C

D

Y

X

E

Z

4) m( CBY)

CONFIDENTIAL 21

Find each angle measure:

1150

Y0K

M

L

5) m( KLM)

6) m( VYX)

Y

X

W Z

(2a+50)0

V 4a0

CONFIDENTIAL 22

State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures:

12

34

5

7) m 1 = (7x+15)0 , m 2 = (10x-9)0

8) m 3 = (23x+15)0 , m 4 = (14x+21)0

CONFIDENTIAL 23

Parallel, perpendicular and skew linesParallel, perpendicular and skew lines

When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed.

1 23 4

5 67 8

There are several special pairs of angles formed from this figure.

Vertical pairs: Angles 1 and 4  Angles 2 and 3  Angles 5 and 8  Angles 6 and 7

Let’s review

CONFIDENTIAL 24

Supplementary pairs:

Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7

1 23 4

5 67 8

Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs.

There are three other special pairs of angles. These pairs are congruent pairs.

CONFIDENTIAL 25

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

Corresponding angle postulateCorresponding angle postulate

1 2 3 45 6 7 8

p q

t

1 3 2 4 5 7 6 8

CONFIDENTIAL 26

Using the Corresponding angle postulateUsing the Corresponding angle postulate

Find each angle measure.

800 x0

B

C

A

A) m( ABC)

x = 80 corresponding angles

m( ABC) = 800

CONFIDENTIAL 27

B) m( DEF)

(2x-45)0 = (x+30)0 corresponding angles

m( DEF) = (x+30)0

(2x-45)0

EF

D

(x+30)0

x – 45 = 30 subtract x from both sides

x = 75 add 45 to both sides

= (75+30)0

= 1050

CONFIDENTIAL 28

Alternate interior angles theoremAlternate interior angles theorem

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

1 3 2 4

1 2

4 3

Theorem

Hypothesis Conclusion

CONFIDENTIAL 29

Alternate exterior angles theoremAlternate exterior angles theorem

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

5 7 6 8

5 6

8 7

Theorem

Hypothesis Conclusion

CONFIDENTIAL 30

Same-side interior angles theoremSame-side interior angles theorem

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

m 1 + m 4 =1800 m 2 + m 3 =1800

Theorem

Hypothesis Conclusion

1 2

4 3

CONFIDENTIAL 31

Alternate interior angles theoremAlternate interior angles theorem

1 2

3m

lGiven: l || m

Prove: 2 3

Proof:

1 3l || m

Given Corresponding angles

2 3

2 1

Vertically opposite angles

CONFIDENTIAL 32

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