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Topic 4.4.1Topic 4.4.1

Parallel LinesParallel Lines

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Topic4.4.1

Parallel LinesParallel Lines

California Standard:8.0 Students understand the concepts of parallel lines and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

What it means for you:You’ll work out the slopes of parallel lines and you’ll test if two lines are parallel.

Key words:• parallel• intersect

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Topic4.4.1

Now that you’ve practiced finding the slope of a line, you can use the method on a special case — parallel lines.

Parallel LinesParallel Lines

Remember the rise over run formula from Topic 4.3.1:

Slope = , provided x 0 vertical change

horizontal change =

rise run

y x

=

You can use it to prove that two lines are parallel.

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Topic4.4.1

Parallel Lines Never Meet

Parallel lines are two or more lines in a plane that never intersect (cross).

Parallel LinesParallel Lines

The symbol || is used to indicate parallel lines — you read this symbol as “is parallel to.” So, if l1 and l2 are lines, then l1 || l2 means “line l1 is parallel to line l2.”

These lines are all parallel. No matter how long you draw them, they’ll never meet.

y-axis

x-axis

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Topic4.4.1

Parallel Lines Have Identical Slopes

You can determine whether lines are parallel by looking at their slopes.

Parallel LinesParallel Lines

Two lines are parallel if their slopes are equal.

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y-axis

x-axis

Topic4.4.1

Prove that the three lines A, B, and C shown on the graph are parallel.

Solution follows…

Example 1

Solution

Parallel LinesParallel Lines

4

2Using the rise over run

formula: slope = ,

you can see that they all

have a slope of = .2

4

1

2

yx

4

2

4

2

A

B

C

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1. Two lines on the same plane that never intersect are called ……………… lines.

2. To determine if two lines are parallel you can look at their ………………

3. Prove that the line f defined by y – 3 = (x – 4) is

parallel to line g defined by y – 6 = (x + 1).

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32

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Topic4.4.1

Guided Practice

Solution follows…

Parallel LinesParallel Lines

parallel

slopes

Lines f and g are both written in point-slope form, so both lines

have slope . The slopes are the same, so the lines are parallel.2

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Topic4.4.1

Vertical Lines Don’t Have Defined Slopes

Vertical lines are parallel, but you can’t include them in the definition on slide 5 because their slopes are undefined.

Parallel LinesParallel Lines

Points on a vertical line all have the same x-coordinate, so they are of the form (c, y1) and (c, y2).

The slope of a vertical line is undefined because

m = = is not defined.y2 – y1

c – c

y2 – y1

0

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Topic4.4.1

Test if Lines are Parallel by Finding Slopes

To check if a pair of lines are parallel, just find the slope of each line.

Parallel LinesParallel Lines

If the slopes are equal, the lines are parallel.

Remember — if you’re given two points on a line, you can find the slope of the line using the formula:

y2 – y1

x2 – x1

m =

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Topic4.4.1

Show that the straight line through (2, –3) and (–5, 1) is parallel to the straight line joining (7, –1) and (0, 3).

Solution follows…

Example 2

Solution

Parallel LinesParallel Lines

Step 2: Compare the slopes and draw a conclusion.

So the straight line through (2, –3) and (–5, 1) is parallel to the straight line through (7, –1) and (0, 3).

1 – (–3)

–5 – 2m1 = = = –

4

–7

4

7

3 – (–1)

0 – 7m2 = = = –

4

–7

4

7

– = – , so m1 = m2.4

7

4

7

Step 1: Find the slope of each line using m =y2 – y1

x2 – x1

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Topic4.4.1

Guided Practice

Solution follows…

Parallel LinesParallel Lines

4. Show that line a, which goes through points (7, 2) and (3, 3), is parallel to line b joining points (–8, –4) and (–4, –5).

5. Show that the line through points (4, 3) and (–1, 3) is parallel to the line though points (–6, –1) and (–8, –1).

Both lines have slopes of zero, so they are parallel.

m1 = = = 03 – 3

–1 – 4

0

–5m2 = = = 0

–1 – (–1)

–8 – (–6)

0

–2

Both lines have the same slope, so they are parallel.

mb = = = ––5 – (–4)

–4 – (–8)

–1

4ma = = = –

3 – 2

3 – 7

1

–4

1

4

1

4

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Topic4.4.1

Guided Practice

Solution follows…

Parallel LinesParallel Lines

Line 1 has an undefined slope (it’s vertical) and line 2 has slope 0 (it’s horizontal). So the lines are not parallel.

6. Determine if line f joining points (1, 4) and (6, 2) is parallel to line g joining points (0, 8) and (10, 4).

Lines f and g both have slopes of – , so the lines are parallel. 2

5

Line 1 has slope and line 2 has slope , so the lines are not parallel. 5

8

5

2

7. Determine if the line through points (–5, 2) and (3, 7) is parallel to the line through points (–5, 1) and (–3, 6).

8. Determine if the line through points (–8, 4) and (–8, 3)

is parallel to the line through points (6, 3) and (–4, 3).

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Topic4.4.1

Find the equation of a line through (–1, 4) that is parallel to the straight line joining (5, 7) and (–6, –8).

Parallel LinesParallel Lines

Example 3

Solution follows…

Solution

Step 1: Find the slope of the line through (5, 7) and (–6, –8).

Step 2: The slope of the line through (–1, 4) must be equal to m1 since the lines are parallel.

Solution continues…

This parallel line problem seems tougher but it’s not too hard.

–8 – 7

–6 – 5m1 = = = –

–15

–11

15

11

So, m1 = m2 = –15

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Topic4.4.1

Find the equation of a line through (–1, 4) that is parallel to the straight line joining (5, 7) and (–6, –8).

Parallel LinesParallel Lines

Example 3

Solution (continued)

Step 3: Now use the point-slope formula to find the

equation of the line through point (–1, 4) with slope .15

11y – y1 = m(x – x1)

15

11 y – 4 = [x – (–1)]

11y – 44 = 15(x + 1)

11y – 44 = 15x + 15 Equation: 11y – 15x = 59

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9. Find the equation of the line through (–3, 7) that is parallel to the line joining points (4, 5) and (–2, –8).

10. Find the equation of the line through (6, –4) that is parallel to the line joining points (–1, 6) and (7, 3).

11. Find the equation of the line through (–1, 7) that is parallel to the line joining points (4, –3) and (8, 6).

Topic4.4.1

Guided Practice

Solution follows…

Parallel LinesParallel Lines

m = = =–8 – 5

–2 – 4

–13

–6

13

6y – 7 = [x – (–3)] 6y – 42 = 13x + 39

6y – 13x = 81

13

6

m = = = –3 – 6

7 – (–1)

–3

8

3

8y – (–4) = – [x – 6] 8y + 32 = –3x + 18

8y + 3x = –14

3

8

m = = 6 – (–3)

8 – 4

9

4y – 7 = [x – (–1)] 4y – 28 = 9x + 9

4y – 9x = 37

9

4

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Topic4.4.1

Guided Practice

Solution follows…

Parallel LinesParallel Lines

12. Write the equation of the line through (–3, 5) that is parallel to the line joining points (–1, 2.5) and (0.5, 1).

13. Write the equation of the line through (–2, –1) that is parallel to the line x + 3y = 6.

m = = = – 11 – 2.5

0.5 – (–1)

–1.5

1.5y – 5 = –1[x – (–3)] y – 5 = –x – 3 y + x = 2

Find two points on the line x + 3y = 6. For example, (0, 2) and (6, 0).Use these points to find m.

m = = – 0 – 2

6 – 0

1

3y – (–1) = – [x – (–2)] 3y + 3 = –x – 2

3y + x = –5

1

3

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Topic4.4.1

Independent Practice

Solution follows…

Parallel LinesParallel Lines

The lines are parallel.

1. Line l1 has slope and line l2 has slope .

What can you conclude about l1 and l2?

3. Show that all horizontal lines are parallel.Points on a horizontal line all have the same y-coordinate, c, where c is a constant. So, for any two points, y2 – y1 = c – c = 0. This means horizontal lines all have a slope of zero, so they are all parallel to each other.

1

2

1

2

2. Line l1 has a slope of – .

If l1 || l2, then what is the slope of l2?

1

3 1

3–

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4. Show that the line through the points (5, –3) and (–8, 1) is parallel to the line through (13, –7) and (–13, 1).

Topic4.4.1

Independent Practice

Solution follows…

Parallel LinesParallel Lines

Line 1 has slope –1 and line 2 has slope – , so the lines are not parallel. 8

5

Both lines have slopes of – , so the lines are parallel. 5

8

m1 = m2, so the lines are parallel.

m1 = = –1 – (–3)

–8 – 5

4

13m2 = = – = –

1 – (–7)

–13 – 13

8

26

4

13

5. Determine if the line through the points (5, 4) and (0, 9) is parallel to the line through (–1, 8) and (4, 0).

6. Determine if the line through the points (–2, 5) and (6, 0) is parallel to the line through (8, –1) and (0, 4).

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Topic4.4.1

Independent Practice

Solution follows…

Parallel LinesParallel Lines

The lines are parallel since they are both vertical lines.

The lines are not parallel, since one is vertical and the other is horizontal.

11y + 8x = –30

10y + 7x = –5

2y + 3x = 12

7. Determine if the line through the points (4, –7) and (4, –4) is parallel to the line through (–5, 1) and (–5, 5).

8. Determine if the line through the points (–2, 3) and (–2, –2) is parallel to the line through (1, 7) and (–6, 7).

9. Find the equation of the line through (1, –2) that is parallel to the line joining the points (–3, –1) and (8, 7).

10. Find the equation of the line through (–5, 3) that is parallel to the line joining the points (–2, 6) and (8, –1).

11. Write the equation of the line through (0, 6) that is parallel to the line 3x + 2y = 6.

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Topic4.4.1

Round UpRound Up

When you draw lines with different slopes on a set of axes, you might not see where they cross.

Parallel LinesParallel Lines

But remember, you are only looking at a tiny bit of the lines — they go on indefinitely in both directions. If they don’t have identical slopes, they’ll cross sooner or later.