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Angles and Parallel Lines

- Geometry D Section 3.2

Angles and Parallel Lines

We are going to investigate the relationship of

various angles created by two parallel lines and

a transversal.

Obtain a ½ sheet of graph paper and a protractor.

Construct two lines and a transversal similar

to the image on the next slide.

Angles and Parallel Lines

Extend your lines the full height and width of

the paper.

Pause for time to work!

Angles and Parallel Lines

Label the angles as shown below.

1

2

3

4

6

5

7

8

Pause for time to work!

Angles and Parallel Lines

Measure all angles using a protractor to the

nearest degree.

1

2

3

4

6

5

7

8

Pause for time to work!

Angles and Parallel Lines

Measure all angles using a protractor to the

nearest degree.

127o

53o

1

2

Note Your measurements may be different values

but should be in the same pattern.

3

4

53o

127o

127o

53o

6

5

7

8

53o

127o

Angles and Parallel Lines

Identify the relationship between the following

angles?

127o

53o

1

2

3

4

53o

127o

127o

53o

6

5

7

8

53o

From Chapter 2, the angles are linear pairs.

127o

What can be said about the measures of the linear

pairs?

Linear pairs are supplementary (sum to 180o).

Angles and Parallel Lines

Identify the relationship between the following

angles?

127o

53o

1

2

3

4

53o

127o

127o

53o

6

5

7

8

53o

From Chapter 2, the angles are vertical angles.

127o

What can be said about the measures of the

vertical angles?

Vertical angles are congruent angles.

Angles and Parallel Lines

Identify the relationship between the following

angles?

127o

53o

1

2

3

4

53o

127o

127o

53o

6

5

7

8

53o

The angles are corresponding angles.

127o

What can be said about the measures of the

corresponding angles?

The measures are equal and the angles are

congruent.

Angles and Parallel Lines

Corresponding Angles Postulate

If two parallel lines are cut by a transversal,

then each pair of corresponding angles are

congruent.

Angles and Parallel Lines

Identify the relationship between the following

angles?

127o

53o

1

2

3

4

53o

127o

127o

53o

6

5

7

8

53o

The angles are alternate interior angles.

127o

What can be said about the measures of the

alternate interior angles?

The measures are equal and the angles are

congruent.

Angles and Parallel Lines

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal,

then each pair of alternate interior angles are

congruent.

You will prove this theorem as a homework problem!

Angles and Parallel Lines

Identify the relationship between the following

angles?

127o

53o

1

2

3

4

53o

127o

127o

53o

6

5

7

8

53o

The angles are alternate interior angles.

127o

What can be said about the measures of the

alternate interior angles?

The measures add to 180o.

Angles and Parallel Lines

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal,

then each pair of consecutive interior angles is

supplementary (sum to 180o).

You will prove this theorem as a homework problem!

Angles and Parallel Lines

Identify the relationship between the following

angles?

127o

53o

1

2

3

4

53o

127o

127o

53o

6

5

7

8

53o

The angles are alternate exterior angles.

127o

What can be said about the measures of the

alternate interior angles?

The measures are equal and the angles are

congruent.

Angles and Parallel Lines

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal,

then each pair of alternate exterior angles is

congruent.

Prove

Statement Reason

?

p q, t is a transversal of p q

Given

t

Corresponding s are

?

1

2

p

3

4

?

6

5

Vertical s are

q

7

8

?

Transitive Property

Angles and Parallel Lines

Perpendicular Transversal Theorem

In a plane, if a line is perpendicular to one of

two perpendicular lines, then it is perpendicular

to the other.

t

p

q

If t is perpendicular ( ) to p, then it is

also perpendicular to q.

You will prove this theorem as a homework problem!

Angles and Parallel Lines

Applications

Gather into groups of not more than 3. Work the

following problems in your group. Compare your

answers to those provided.

Angles and Parallel Lines

Given j k,

Applications

Make a sketch of the problem in your notes.

Find the measure of

4

5

3

1.

43o

2

1

7

8

6

Corresponds with 1.

9

11

10

2.

24o

12

Alternate exterior with 14.

13

3.

156o

Linear pair with 9. 180o 24o 156o

14

Angles and Parallel Lines

Given j k, Find the measure of

Applications

4

5

3

4.

137o

2

1

7

8

6

Linear pair with 3.

9

11

10

5.

156o

12

Vertical angle with 10.

13

6.

43o

Vertical with 1. Alternate Interior of

3.

14

Angles and Parallel Lines

Applications

Find the values of x and y in each figure. Find

the measure of each given angle. Note Figures

are not drawn to scale.

Given

Pause for time to work!

Angles and Parallel Lines

Applications Solution

Given

Linear pairs are supplementary.

(5x 2) (9x 10) 180o

14x 12 180

14x 168

x 12

Linear Pair

By corresponding angles,

62o

118o

3y 1 62

62o

3y 63

y 21 and

Angles and Parallel Lines

Applications

Find the values of x, y and z in each figure.

(2z)o

(3x3)o

(4y2)o

66o

Pause for time to work!

Angles and Parallel Lines

is a corresponding angle with the

angle of 66o.

Applications

(3x 3)o and 66o are linear pairs and sum to

180o.

(3x 3)o 66o 180o 3x 63 180 3x 117,

x 39

(2z)o

(3x3)o

66o

(4y 2)o and 66o are congruent alternate

interior angles.

(4y 2)o 66o 4y 64, y 16

(4y2)o

66o

(3x3)o and (2z)o are congruent alternate

interior angles.

(3x3)o 3(39) 3 114o

(2z)o 114o, z 57

Angles and Parallel Lines

Applications

Find the values of x, y and z in each figure.

There are other ways of doing this problem

correctly. If you worked it a different way,

would you be willing to share how you did it?

(2z)o

(3x3)o

(4y2)o

66o

Angles and Parallel Lines

Applications

Find the measures of all the angles on the object

if the measure of angle 1 is 30o.

Pause for time to work!

Angles and Parallel Lines

Applications

Find the measures of all the angles on the object

if the measure of angle 1 is 30o.

Perpendicular transversal theorem.

Perpendicular lines intersect in 4 right (90o)

angles.

90o

90o

90o

90o

90o

Angles and Parallel Lines

Applications

Find the measures of all the angles on the object

if the measure of angle 1 is 30o.

Vertical Angle

Vertical Angle

Alternate interior angle with angle 1.

30o

150o

150o

30o

90o

30o

Linear pairs are supplementary.

90o

30o

90o

Given

90o

90o

Vertical Angles

Angles and Parallel Lines

Applications

Find the measures of all the angles on the object

if the measure of angle 1 is 30o.

30o

Vertical angles.

150o

150o

30o

30o

90o

60o

60o

All angles have been found!

90o

30o

90o

90o

Since the transversal is , these two angles

must add to 90o using angle addition.

90o

Angles and Parallel Lines

Assignment

3.2 / 17-20, 24, 26, 30, 32, 36, 38, 40,

43, 48, 55, 57, 60, 62

Please return your protractor!!!! Thank you Mr.

Matzke!!!!