Parallelism Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parallelism.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Lines in the same plane that never intersect because they maintain a constant distance from each other.

Railroad tracks—they stay exactly the same distance apart and never meet, no matter how far they extend.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Parallel lines run in the exact same direction, staying a constant distance apart so they never cross.

Common stuck point: The procedure for parallelism is the easy part; the trap is confusing equal slopes with negative-reciprocal slopes. Asking "Do the two lines have exactly equal slopes so they never meet?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the two lines have exactly equal slopes so they never meet?

Worked Examples

Example 1

easy

Line

\ell_1

passes through

(0, 2)

and

(4, 6)

. Write the equation of a line

\ell_2

parallel to

\ell_1

passing through

(1, -3)

Answer

y = x - 4

First step

Step 1: Slope of

\ell_1

m_1 = \dfrac{6-2}{4-0} = 1

Full solution

2
Step 2: Parallel lines have equal slopes, so $m_2 = 1$ .
3
Step 3: Point-slope form through $(1, -3)$ : $y + 3 = 1(x - 1) \Rightarrow y = x - 4$ .

Two distinct lines in the same plane are parallel if and only if they have equal slopes. Here both lines have slope

1

but different

y

-intercepts (

2

and

-4

), confirming they never intersect.

Example 2

medium

Transversal

t

crosses parallel lines

\ell_1 \parallel \ell_2

. If the co-interior (same-side interior) angle at

\ell_1

65°

, find the co-interior angle at

\ell_2

and the alternate interior angle at

\ell_2

Example 3

easy

Find the equation of the line through

(2, 1)

parallel to

y = -x + 8

Example 4

medium

A transversal cuts two parallel lines. One pair of corresponding angles measures

4x + 10

and

6x - 20

. Find

x

Example 5

medium

\ell_1 \parallel \ell_2

and a transversal makes an alternate-interior angle of

72°

with

\ell_1

, what is the co-interior (same-side interior) angle with

\ell_2

Example 6

medium

Write the equation of the line parallel to

3x - 2y = 6

that passes through

(4, -1)

Example 7

medium

Show whether the line through

(1, 1)

and

(4, 7)

is parallel to the line through

(0, -2)

and

(2, 2)

Example 8

hard

Find the distance between the parallel lines

y = 2x + 3

and

y = 2x - 7

Example 9

hard

\triangle ABC

D

is on

AB

with

AD = 3

DB = 6

, and

E

is on

AC

such that

DE \parallel BC

. If

AC = 15

, find

AE

Example 10

hard

Two parallel lines

y = mx

and

y = mx + c

(

c > 0

) are cut by the

x

-axis at

A

and

B

respectively. Express

AB

in terms of

m

and

c

Example 11

challenge

Prove that the midpoints of the sides of any quadrilateral form a parallelogram (Varignon's theorem).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Are the lines

y = 3x + 7

and

6x - 2y = 4

parallel? Justify your answer.

Example 2

hard

Parallelogram

PQRS

has

P(0,0)

Q(5,0)

R(7,4)

. Find coordinates of

S

so that

PQ \parallel SR

and

PS \parallel QR

Example 3

easy

Do parallel lines ever intersect?

Example 4

easy

Two lines have slopes 3 and 3. Are they parallel?

Example 5

easy

Is the line

y = 2x + 1

parallel to

y = 2x + 5

Example 6

easy

Are two vertical lines parallel?

Example 7

easy

What stays constant between two parallel lines?

Example 8

easy

Line A has slope 4. For a line to be parallel to A, what must its slope be?

Example 9

easy

Are the lines

y = 2x + 1

and

y = 3x + 1

parallel?

Example 10

easy

In a parallelogram, how many pairs of parallel sides are there?

Example 11

medium

Find the slope of a line parallel to the line through

(1, 2)

and

(5, 10)

Example 12

medium

Write the equation of the line through

(0, 5)

parallel to

y = 3x - 2

Example 13

medium

Why can't you trust that two lines drawn 'looking parallel' in a diagram are actually parallel?

Example 14

medium

A transversal crosses two parallel lines. Corresponding angles are

3x

and

75^\circ

. Find

x

Example 15

medium

A transversal crosses two parallel lines making co-interior (same-side interior) angles of

x

and

110^\circ

. Find

x

Example 16

medium

Two lines are each parallel to a third line. Are the two lines parallel to each other?

Example 17

medium

Is the line

2y = 6x + 4

parallel to

y = 3x - 7

Example 18

medium

Does a translation map a line to a line parallel to the original?

Example 19

challenge

Find the distance between the parallel lines

y = 2x + 1

and

y = 2x + 6

Example 20

challenge

A transversal makes alternate interior angles of

(2x + 10)^\circ

and

(3x - 20)^\circ

with two lines. For the lines to be parallel, find

x

Example 21

challenge

In a trapezoid with parallel sides of lengths 6 and 10, a midsegment connects the midpoints of the non-parallel sides. Find its length.

Example 22

challenge

Explain why two distinct lines in a plane are parallel if and only if they have no point in common, but this fails in 3D space.

Example 23

easy

What is the slope of any line parallel to

y = -4x + 9

Example 24

easy

Are the lines

y = \tfrac{1}{2}x + 3

and

y = \tfrac{1}{2}x - 7

parallel?

Example 25

easy

Write the equation of the line parallel to

y = 2x + 5

that passes through

(0, -3)

Example 26

easy

Identify which pair of lines is parallel: (a)

y = 3x + 1

and

y = -3x + 1

, (b)

y = 5x - 2

and

10x - 2y = 7

Example 27

medium

Two lines

\ell_1: 2x + 3y = 12

and

\ell_2: 4x + 6y = 5

. Are they parallel?

Example 28

medium

Find the value of

k

so that the lines

y = (k+1)x + 2

and

y = (3k - 5)x - 4

are parallel.

Example 29

medium

A pair of parallel lines is cut by a transversal. The two interior angles on the same side of the transversal measure

3y

and

5y + 20

. Find

y

Example 30

medium

A parallelogram has vertices

A(0, 0)

B(6, 0)

D(2, 3)

. Find

C

so that

ABCD

is a parallelogram.

Example 31

medium

A line passes through

(-1, 4)

and is parallel to

5x + y = 9

. Write its equation in slope-intercept form.

Example 32

hard

For what value of

a

are the lines

ax + 2y = 5

and

6x + (a + 1)y = 9

parallel but not identical?

Example 33

hard

Quadrilateral

WXYZ

has

W(0, 0)

X(7, 2)

Y(9, 8)

Z(2, 6)

. Is it a parallelogram?

Example 34

hard

A line parallel to

y = -\tfrac{3}{4}x + 2

passes through the point of intersection of

x + y = 5

and

x - y = 1

. Find its equation.

← Back to Parallelism Formula Explained Practice Mode

Related Concepts

Line Slope Transversal Angles

Background Knowledge

These ideas may be useful before you work through the harder examples.

lineslope