Topic 1Parallel Lines

Unit 2 Topic 1

Explore: Angles formed by Parallel Lines

• Take a few minutes to watch the following video.

• http://www.khanacademy.org/math/geometry/angles/v/angles--part-3

You can scan the QR code to the right if you have a SmartPhone. Simply download a scanner app like RedLaser.

Note: Parallel lines are lines that never meet.

Information

Adjacent angles are pairs of angles that share a common vertex and common side. Two special types of adjacent angles are described below.• Complementary angles are two angles with a sum of 90°. • Supplementary Angles are two angles with a sum of 180°. • Straight Line Angles are angles along a straight line with a

sum of 180°.

Perpendicular lines cross each other at right angles. They can be marked with one right angle symbol at the intersection point of the lines. The two lines in the figure shown on the right are perpendicular.

Example 1Finding missing angles

In each figure, find the specified angle. a) b)

c) d)

Try this on your own first!!!!

Example 1: SolutionsFinding missing angles

In each figure, find the specified angle. a) b)

c) d)

90 65

25

180 38

142

3056

90 2

45

86 30 56

Information

• Opposite angles are formed when two lines cross each other. The measures of opposite angles are equal. In the figure shown on the right, a = b and c = d.

• Parallel lines do not cross each other. They can be marked by matching arrowheads located along the line. A transversal is a line that crosses two or more parallel lines. In the figure shown on the right,

lines and are parallel. Line is a transversal.

1l 3l2l

Information

•Corresponding angels are equal.

•Alternate interior angles are equal.

•Same side interior angles add to 180°.

x

x

x

x

xx

Example 2Identifying Angles Formed by Two Parallel Lines

Identify the following pairs of angles shown in the figure on the right.a) opposite angles

b) alternate interior angles

c) corresponding angles

d) same side interior angles

Try this on your own first!!!!

123 4

5 68 7

Example 2: SolutionIdentifying Angles Formed by Two Parallel Lines

Identify the following pairs of angles shown in the figure on the right.a) opposite angles

b) alternate interior angles

c) corresponding angles

d) same side interior angles

1 3 5 7

2 4 6 8

12

3 4

5 68 73 6

4 5

1 6 4 7

2 5 3 8

3 5 180

4 6 180

Example 3Determining if Two Lines Are Parallel

In each figure, determine if lines AB and CD are parallel with each other. Explain why or why not.a) b)

c)

Try this on your own first!!!!

Example 3: SolutionDetermining if Two Lines Are Parallel

a) b)

c)

Parallel because the alternate interior angles are equal.

40 40 Not parallel because the same side interior angles do not sum to 180°.102 70 172

Parallel because opposites angles are equal and then corresponding angles are equal.

85

85 85

Example 4aDetermining Missing Angles

Determine the measures of the missing angles in each of the diagrams.

Try this on your own first!!!!

67A

F

B

D E

C

G

Angle Measure

A =

B =

C =

D =

E =

F =

G =

67A

F

B

D E

C

G

Angle Measure

Reason

A = 113⁰Angles that lie beside each other are supplementary. (180-67=113).

B = 67⁰ Opposite angles are equal.

C = 113⁰ Opposite angles are equal.

D = 113⁰Alternate interior angles are equal (C=D).

E = 67⁰Interior angles are supplementary (E=180-C). 180-113=67.

F = 67⁰Alternate exterior angles are equal.

G = 113⁰Corresponding angles are equal. (C=G)

Example 4a: Solution

113⁰

67⁰ 113⁰

113⁰ 67⁰

67⁰ 113⁰

Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.

Example 4b Try this on your own first!!!!

138

AB

CD

EF G

Angle Measure

A =

B =

C =

D =

E =

F =

G =

Example 4b: Solution

Example has a video solution. Click here!

Note: The solution has also been provided on the following slide.

Example 4b: Solution

138

AB

CD

EF G

Angle Measure

Reason

A = 42 Corresponding angles are equal. (A = E)

B = 138 Straight angle. (A +B = 180)

C =138 Same side interior angles.(C +E = 180)

D = 42 Alternate interior angles are equal. (D = E )

E = 42Straight angle (*started here).(E +138 = 180)

F = 138 Opposite angles are equal.(F = 138)

G = 42 Opposite angles are equal.(G = E)

138 4242

4242

138138

Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.

Example 4c Try this on your own first!!!!

ab

c

d110

Angle Measure

a =

b =

c =

d =

Example 4c: Solution

ab

c

d110

Angle Measure

Reason

a = 110 Corresponding angles are equal.

b = 110 Opposite angles are equal.

c = 70 Interior angles are supplementary.

d = 70 Alternate interior angles are equal.

110

110

70

70

Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.

Example 4d Try this on your own first!!!!

Angle Measure

w =

x =

y =

w

x

y 120

Example 4d: Solution

Angle Measure

Reason

w =120 Opposite angles are equal.

x = 60 Interior angles are supplementary.

y = 60 Corresponding angles are equal.

w

x

y 120

120

60

60

Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.

Example 4e Try this on your own first!!!!

Angle Measure

a =

b =

c =

d =

e =

f =

55

112

a

e

d

f

cb

Example 4e: Solution

Example has a video solution. Click here!

Note: The solution has also been provided on the following slide.

Example 4e: Solution

55

112

a

e

d

f

cb

Angle Measure

Reason

a = 112 Corresponding angles are equal. (a = 112)

b = 55 Opposite angles are equal. (b =f)

c =68 Straight angle.(c + 112 = 180)

d = 55 Opposite angles are equal. (d = 55 )

e = 112 Alternate interior angles are equal. (e = 112)

f = 55 Alternate interior angles are equal. (f = d)

Note: Your explanation for each angle measurement may differ from those here. There are many explanations for each measure.

112

112

55

55

5568

Example 5Proving equal angles in parallel lines

Given that lines k and m are parallel, prove that

Try this on your own first!!!!

A C

A

CB

k

m

Statement Reason

Example 5: Solution

A

CB

k

m

Statement Reason

Given

B Corresponding angles are equal.

Opposite angles are equal.

Deductive conclusion. (Both equal .)

Need to Know:• When a transversal intersects parallel lines, several

relationships are formed.

• Opposite angles are equal angles, directly across from each other.

• Corresponding angles are equal angles, where one interior angle and one exterior angle are on the same side of the transversal (F rule).

• Alternate interior angles are equal angles, where two interior angles are on opposite sides of the transversal (Z rule).

Need to Know:• Interior angles are supplementary angles, where

the two interior angles are on the same side of the transversal. (C rule).

• Supplementary angles are angles that have a sum of 180.

• Alternate exterior angles are equal angles, where the two exterior angles are on the opposite sides of the transversal.

You’re ready! Try the homework from this section.