Angles and Parallel Lines Geometry D – Section 3.2

Angles and Parallel Lines

Geometry D – Section 3.2


Obtain a ½ sheet of graph paper and a protractor. Construct two || lines and a transversal similar to the image on the next slide.

We are going to investigate the relationship of various angles created by two parallel lines and a transversal.


Extend your lines the full height and width of the paper.

Pause for time to work!


Label the angles as shown below.

1 2

3 4

657 8



Measure all angles using a protractor to the nearest degree.

1 2

3 4

657 8



Measure all angles using a protractor to the nearest degree.

1 2

3 4

657 8

127o

Note: Your measurements may be different values but should be in the same pattern.

127o

127o

127o

53o

53o

53o

53o

Angles and Parallel LinesIdentify the relationship between the following angles?

1 2

3 4

657 8

127o

From Chapter 2, the angles are linear pairs.

127o

127o

127o

53o

53o

53o

53o

1, 2

3, 4

5, 6

7, 8

What can be said about the measures of the linear pairs?

Linear pairs are supplementary (sum to 180o).


1 2

3 4

657 8

127o

From Chapter 2, the angles are vertical angles.

127o

127o

127o

53o

53o

53o

53o

1, 4

2, 3

5, 8

6, 7

What can be said about the measures of the vertical angles? Vertical angles are congruent angles.


1 2

3 4

657 8

127o

The angles are corresponding angles.

127o

127o

127o

53o

53o

53o

53o

1, 5

2, 6

3, 7

4, 8

What can be said about the measures of the corresponding angles? The measures are equal and the angles are

congruent.


Corresponding Angles Postulate –

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


Identify the relationship between the following angles?

1 2

3 4

657 8

127o

The angles are alternate interior angles.

127o

127o

127o

53o

53o

53o

53o

3, 6

4, 5

What can be said about the measures of the alternate interior angles? The measures are equal and the angles are

congruent.


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!



1 2

3 4

657 8

127o

The angles are alternate interior angles.

127o

127o

127o

53o

53o

53o

53o

3, 5

4, 6

What can be said about the measures of the alternate interior angles? The measures add to 180o.


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).




1 2

3 4

657 8

127o

The angles are alternate exterior angles.

127o

127o

127o

53o

53o

53o

53o

1, 8

2, 7

What can be said about the measures of the alternate interior angles? The measures are equal and the angles are

congruent.

1 8, 2 7 Statement Reason


1 5, 2 6

5 8, 6 7

1 8, 2 7

Alternate Exterior Angles Theorem –

If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

t

p

q

1 23 4

65

7 8

p || q, t is a transversal of p & q

Given

Prove:

Corresponding ‘s are

Vertical ‘s are Transitive Property

?

?

?

?


Perpendicular Transversal Theorem –

In a plane, if a line is perpendicular to one of two perpendicular lines, then it is perpendicular to the other.


p

q

t

If t is perpendicular ( ) to p, then it is also perpendicular to q.


Applications –

Gather into groups of not more than 3.

Work the following problems in your group.Compare your answers to those provided.

Make a sketch of the problem in your notes.

Given j || k, Angles and Parallel Lines

jk

Applications –

1 2 3 4 5

10

6 7 8 911

1213

14

1 43 , 14 24o om m

3 1.

Corresponds with 1.

43o

9 2.Alternate exterior with 14.

24o

10 3.Linear pair with 9.180o – 24o = 156o

156o

Find the measure of

jk

Given j || k,

Find the measure of


Applications –

1 2 3 4 5

10

6 7 8 911

1213

14

1 43 , 14 24o om m

4 4.

Linear pair with 3.

137o

11 5.Vertical angle with 10.

156o

7 6.Vertical with 1.Alternate Interior of 3.

43o

D

C A

EB

Find the values of x and y in each figure.Find the measure of each given angle.

Note: Figures are not drawn to scale.


(9 10)om ACB x

Applications –

(5 2)om DCA x Given:

(3 1)om DBE y


D

C A

EB


Applications – Solution

(5 2)om DCA x Given:

(9 10)om ACB x (3 1)om DBE y

180om DCA m ACB Linear pairs are supplementary.

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

14x + 12 = 18014x = 168x = 12

5(12) 2 62om DCA 9(12) 10 118om ACB

By corresponding angles,m DBE m DCA 3y – 1 = 623y = 63 y = 21 and 62om DBE

LinearPair

62o

118o

62o

D

A

B

C

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


Applications –

66o

(3x–3)o(2z)o

(4y+2)o


D

A

B

C

is a corresponding angle with the angle of 66o.Angles and Parallel Lines

Applications –

ABC

66o

(3x–3)o(2z)o

(4y+2)o

66o

(3x – 3)o and 66o are linear pairs and sum to 180o.(3x – 3)o + 66o = 180o

3x + 63 = 1803x = 117, x = 39

(4y + 2)o and 66o are congruent alternate interior angles.

(4y + 2)o = 66o 4y = 64, y = 16(3x–3)o and (2z)o are congruent alternate interior angles.(3x–3)o = 3(39) – 3 = 114o

(2z)o = 114o, z = 57

D

A

B

C

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


Applications –

66o

(3x–3)o(2z)o

(4y+2)o

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?

1


Find the measures of all the angles on the object if the measure of angle 1 is 30o.

Applications –


1



Applications –

90o

90o90o

Perpendicular lines intersect in 4 right (90o) angles. 90o

90o

Perpendicular transversal theorem.

1Linear pairsare supplementary.



Applications –

30o

90o90oGiven

90o

90o

30o

Vertical Angle

90o

150o

150o

VerticalAngle

Alternate interior angle with angle 1.

30o

30o

VerticalAngles

1



Applications –

30o

90o90o

90o

90o

30o

90o

150o

150o

30o

30o

Since the transversal is , these two angles must add to 90o using angle addition.

60o60o

Vertical angles.

All angles have been found!


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!!!!

Documents

Angles and Parallel Lines Geometry D – Section 3.2