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Chapter4
Lesson 1: Slope and Equations of Lines
Every line has a slope. If the line slopes upward, the slope is positive.
+
If the line slopes downward, the slope is negative.
–
If the line is horizontal, the slope is zero.
0
If the line is vertical, the slope is undefined.
Ø
Slope equals the rise divided by the run. The rise is the change in the y coordinates. The run is the
change in the x coordinates.
2
Slope =
=
=
1
This line has a slope of ½ because
it goes up 1 unit every time it
goes over 2 units.
b
Parallel lines have equal slopes. Lines a and b both have a slope of 2/3.
a
The slopes of perpendicular lines are negative reciprocals.
For example, Line a has a slope of 2/3. Line c is perpendicular
to Line a and it has a slope of -3/2.
c
The equation of a line can be written in slope-intercept form. m represents the slope of the line. b
represents the y-intercept of the line. The y-intercept is the point where the line crosses the y-axis.
Slope-Intercept Form:
y = mx + b
For the line y = ¼ x – 5, the slope of the line is ¼ and its y-intercept is -5.
Another useful way to write the equation of a line is the point-slope form. m represents the slope
of the line. x1 and y1 represents the coordinates of a point on the line .
Point-Slope Form:
y – y1 = m(x – x1)
The equation of the line that goes through (-4, 7) and has a slope of 2 would look like this:
y – 7 = 2(x + 4)
If you want to change a point-slope equation into a slope-intercept equation, just distribute the
slope and then add or subtract to get the y by itself.
y – 7 = 2(x + 4)
y – 7 = 2x + 8
y = 2x + 15
Distribute the slope.
Add 7 to both sides.
Horizontal lines have a slope of zero, so their equations can be simplified to y = y1.
y1 is the y-coordinate of any point on the line. The equation of a horizontal line that passes through
(11, 17) is y = 17.
Vertical lines have undefined slope, so they need a special equation that doesn’t contain m. The
equation for a vertical line is x = x1. x1 is the x-coordinate of any point on the line. The equation of
the vertical line that passes through (6, 13) is x = 6.
Chapter 4, Lesson 1
Find the slope between the points.
1. (3, 5) (7, 8)
2. (-1, 2) (4, -10)
3. (6,-3) (-8, -5)
Find the slope perpendicular to the given slope.
4. ¾
5. -4
6. 3
Tell whether the lines through the given points are parallel, perpendicular, or just intersecting.
7. Line 1: (1, 0) (7, 4)
Line 2: (7, 0) (3, 6)
8. Line 1: (-3, 1) (-7, -2)
Line 2: (2, -1) (8, 4)
9. Line 1: (-9, 3) (-5, 7)
Line 2: (-11, 6) (-7, 2)
10. A line has a slope of 2 and goes through these points: (-3, 2) and (0, y). Find the value of y.
Write the equation of each line using the slope-intercept form.
11.
12.
13.
14. A line has the equation 3x + 4y = 24.
a. Transform the equation into slope-intercept form.
b. What is the slope of the line?
c. What is its y-intercept?
15. A line passes through (-5, 1) and has a slope of ½.
a. Write its equation in point-slope form.
b. Transform the equation to slope-intercept form.
c. What is the slope of the line?
d. What is its y-intercept?
16. A line passes through the points (4, 4) and (-8, -5).
a. What is the slope between these points?
b. Write an equation of the line using point-slope form.
c. Transform the equation to slope-intercept form.
d. What is the slope of the line?
e. What is its y-intercept?
17. A line passes through the point (-2, 6) and is parallel to the line y = -4x + 5. Write an equation
of the line in point-slope form.
18. A line passes through the point (7, -1) and is perpendicular to the line y = 2/3x – 8. Write an
equation of the line in point-slope form.
Write the equation of the horizontal line that passes through each of the given points.
19. (6, 9)
20. (-3, 4)
21. (0, -8)
Write the equation of the vertical line that passes through each of the given points.
22. (6, 9)
23. (-3, 4)
24. (0, -8)
BONUS:
Lesson 2: Arrangements of Lines
Two lines in the same plane divide the plane into three different regions. The region between both
lines is called the interior region. The outside areas are called the exterior regions.
exterior
interior
exterior
Two lines are in the same plane will either be intersecting or parallel. Intersecting lines have
different slopes; parallel lines have the same slope.
c
a
b
d
Intersecting Lines
a || b
Parallel Lines
c ||d
A line that intersects two or more lines is called a transversal. In the figure below, line v is a
transversal. When a transversal cuts across two lines, it creates four angles at each intersection.
Angles 3, 4, 5, and 6 are interior angles. Angles 1, 2, 7, and 8 are exterior angles.
v
1 2
3 4
5
7
6
8
These angles can be group in four different pairs.
Corresponding angles – “Corresponding” means “matching”. Corresponding angles
have matching positions at each intersection. Angle 1 is the top, left angle at the
top intersection and Angle 5 is the top, left angle at the bottom intersection, so
Angles 1 and 5 are corresponding angles. Other pairs of corresponding angles are 2
and 6, 3 and 7, and 4 and 8.
Alternate Interior Angles – These angles are located between the two lines in the
interior region and they are on opposite lines and opposites sides of the transversal.
Angle 3 and Angle 6 are alternating interior angles. Angles 4 and 5 are also alternate
interior angles.
Alternate Exterior Angles – These angles are located outside the two lines in the
exterior region and they are on opposite lines and opposites sides of the
transversal. Angle 1 and Angle 8 are alternating exterior angles. Angles 2 and 7
are also alternate exterior angles.
Same Side Interior Angles – These angles are located between the two lines in
the interior region and they are on opposite lines, but the same side of the
transversal. Angle 3 and 5 are same side interior angles. Angles 4 and 6 are also
same side interior angles.
Chapter 4, Lesson 2:
25. If two lines are in the same plane, what are the two possible ways they can be arranged?
26. What is a transversal?
Use the diagram on the right for Questions 26 - 29.
27. Identify all pairs of corresponding angles.
28. Identify all pairs of alternate interior angles.
29. Identify all pairs of alternate exterior angles.
29. Identify all pairs of same side interior angles.
l
i
k
e
m
a
t
h
Use the diagram below for Questions 30 – 36.
1 2
5 6
9 10
13
14
3
7
4
8
11 12
15
16
Classify the angle pairs as corresponding, alternate interior, alternate exterior, or same side interior
angles.
30. ∠5 and ∠13
34. ∠3 and ∠11
31. ∠6 and ∠7
35. ∠14 and ∠15
32. ∠8 and ∠11
36. ∠6 and ∠9
33. ∠13 and ∠12
Lesson 3: Properties of Parallel Lines
Although it seems logical, this postulate has never been proven.
Parallel Postulate – Through a point not on a line, there exists exactly one line parallel to
the given line.
C .
Construction #6: Parallel Line through a Point
1. Draw a line through point C that crosses line a.
a
2. Put the needle of the compass at B and swing an arc
across both lines.
C .
B
3. Do not adjust the compass. Put the needle of the
compass at C and swing a matching arc.
C .
B
4. Use the compass to measure the distance between D and E.
Put the needle at D and adjust the compass to reach E.
D
E
5. Transfer this measurement to the second arc.
C .
6. Use a straightedge to draw a line that goes through
the intersection of the arcs and the given point.
.
C .
These theorems and their converses deal with parallel lines. If you need to prove that two
lines are parallel, use one of the theorems on the left. If you know that the lines are parallel
and you want to show a relationship between two angles, use the theorems on the right.
Use these theorems to prove
that two lines are parallel.
Use these theorems to show a
relationship between angles.
If the corresponding angles are
equal, then the lines are
parallel.
If the lines are parallel, then the
corresponding angles are
equal.
If the alternate interior angles
are equal, then the lines are
parallel.
If the lines are parallel, then the
alternate interior angles are
equal.
If the alternate exterior angles
are equal, then the lines are
parallel.
If the lines are parallel, then the
alternate exterior angles are
equal.
If the same side interior angles
are supplementary, then the
lines are parallel.
If the lines are parallel, then the
same side interior angles are
supplementary.
In a plane if two lines are
perpendicular to a third line,
then they are parallel to each
other.