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Parallel Lines

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Definition

• Two lines are parallel if they lie in the same plane and do not intersect.

• If lines m and n are parallel we write.m n

m n

q

p

m n

p and q are not parallel

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• We also say that two line segments, two rays, a line segment and a ray, etc. are parallel if they are parts of parallel lines.

m

A BC

D

P

Q

AB m is not parallel to PQCD

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Transversals• A transversal for lines m and n is a line t

that intersects lines m and n at distinct points. We say that t cuts m and n.

n

m

t

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• When a transversal cuts parallel lines, special pairs of angles are formed that are sometimes congruent and sometimes supplementary.

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Corresponding Angles• A transversal creates two groups of four

angles in each group. Corresponding angles are two angles, one in each group, in the same relative position.

1 23 4

5 67 8

1 and 5 are corresponding 2 and 6 are corresponding 3 and 7 are corresponding 4 and 8 are corresponding

m

n

If , then corresponding

angles are congruent.

m n

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Vertical Angles• When a transversal cuts two lines, vertical

angles are angles that are opposite one another and share a common vertex.

1

4

3

2

m

n Angles 1 and 3 are vertical angles.

Angles 2 and 4 are vertical angles.

If m || n, then vertical angles are congruent.

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Alternate Interior Angles• When a transversal cuts two lines,

alternate interior angles are angles within the two lines on alternate sides of the transversal.

1

2

3

4

m

n

1 and 2 are alternate interior angles 3 and 4 are alternate interior angles

If , then the alternate

interior angles are congruent.

m n

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Alternate Exterior Angles• When a transversal cuts two lines,

alternate exterior angles are angles outside of the two lines on alternate sides of the transversal.

m

n

1

2

3

4

1 and 2 are alternate exterior angles

3 and 4 are alternate exterior angles

If then the alternate

exterior angles are congruent.

m n

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Interior Angles on the Same Sideof the Transversal

• When a transversal cuts two lines, interior angles on the same side of the transversal are angles within the two lines on the same side of the transversal.

m

n

1

2

3

4

1 and 2 are interior angles on

the same side of the transversal

3 and 4 are interior angles on

the same side of the transversal

If then the interior angles on the same

side of the transversal are supplementary.

m n

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Exterior Angles on the Same Sideof the Transversal

• When a transversal cuts two lines, exterior angles on the same side of the transversal are angles outside of the two lines on the same side of the transversal.

m

n

1

2

3

4

1 and 2 are exterior angles on

the same side of the transversal

3 and 4 are exterior angles on

the same side of the transversal

If then the exterior angles on the same

side of the transversal are supplementary.

m n

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Example

• In the figure and• Find• Since angles 1 and 2 are vertical, they are

congruent. So,• Since angles 1 and 3 are corresponding angles, they are congruent. So,

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3

m n

m n 1 120 .m 2 and 3.m m

2 120 .m

3 120 .m

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Example

• In the figure, and • Find• Consider as a transversal for the parallel line segments.• Then angles B and D are alternate interior angles and so they are congruent. So,

A B

C

D E

40 and 60m A m B

.m D.AB DE

BD

60 .m D

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Example

• In the figure, and• If then find• Considering as a transversal, we see that

angles A and B are interior angles on the same side of the transversal and so they are supplementary.

• So,• Considering as a transversal, we see that

angles B and D are interior angles on the same side of the transversal and so they are supplementary.

• So,

A B

C D

AB CD .AC BD89m A .m D

89

?

AB

180 89 91 .m B

91

89BD

180 91 89 .m D

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Example

• In the figure, bisects

and• Find• Note that is twice• So,• Considering as a transversal for the

parallel line segments, we see that

are corresponding angles and so they are congruent.

• So,

A

B C

D E

BE ,ABC 40 ,m DBE .DE BC.m ADE

m ABC .m DBE80 .m ABC BD

and ADE ABC

80 .m ADE

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Example

• In the figure is more than and is less than twice

• Also, Find• Let denote Then

• Note that angles 2 and 4 are alternate interior angles and so they are congruent.

• So, Adding 44 and subtracting from both sides gives

• So, Note that angles 1 and 5 are alternate interior angles, and so

1

2

34

5

4m 2 1m2m 44 1.m

3.mx 1.m

4 2 and 2 2 44 .m x m x

mn.m n

2 44 2.x x x 46 .x

1 46 and 4 48 .m m

5 46 .m Now 3 180 46 48 86 .m

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Proving Lines Parallel

• So, far we have discussed that if we have a pair of parallel lines, then certain pairs of angles created by a transversal are congruent or supplementary.

• Now we consider the converse. • If two lines are cut by a transversal so that

corresponding angles are congruent, then the lines are parallel.

• If the alternate interior or exterior angles are congruent, then the lines are parallel.

• If the interior or exterior angles on the same side of the transversal are supplementary, then the lines are parallel.

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Example

• In the figure, angles A and B are right angles and

• What is• Since these

angles are supplementary. Note that they are interior angles on the same side of the transversal This means that

• Now, since angles C and D are interior angles on the same side of the transversal they are supplementary.

• So,

A

B C

D

78 .m C ?m D90 and 90 ,m A m B

.ABAD BC

,CD180 78 102 .m D