M = (
ᎏ
–3
2
+1
ᎏ
,
ᎏ
5+
2
(–4)
ᎏ
)
M = (–
ᎏ
2
2
ᎏ
,
ᎏ
1
2
ᎏ
)
M = (–1,
ᎏ
1
2
ᎏ
)
Note: There is no such thing as the midpoint of a line, as lines are infinite in length.

SLOPE
The slope of a line (or line segment) is a numerical value given to show how steep a line is. A line or segment can
have one of four types of slope:
■
A line with a positive slope increases from the bottom left to the upper right on a graph.
■
A line with a negative slope decreases from the upper left to the bottom right on a graph.
■
A horizontal line is said to have a zero slope.
■
A vertical line is said to have no slope (undefined).
■
Parallel lines have equal slopes.
■
Perpendicular lines have slopes that are negative reciprocals of each other.
Positive slope
y
1
4
3
2
–5
–1
–2
–3
–4
1
5432
–5
–1

–2
–3–4
x
5
y
1
4
3
2
–5
–1
–2
–3
–4
1
5432
–5
–1
–2
–3–4
x
5
y
1
4
2
–5
–1
–2
–3

–4
1
5432
–5
–1
–2
–3–4
x
5
y
1
4
3
2
–5
–1
–2
–3
–4
1
542
–5
–1
–2
–3–4
x
5
3
3
Negative slope

Zero slope
Undefined (no) slope
– THEA MATH REVIEW–
145
The slope of a line can be found if you know the coordinates of any two points that lie on the line. It does
not matter which two points you use. It is found by writing the change in the y-coordinates of any two points on
the line, over the change in the corresponding x-coordinates. (This is also known as the rise over the run.)
The formula for the slope of a line (or line segment) containing points (x
1
, y
1
) and (x
2
, y
2
): m =
ᎏ
y
x
2
2
–
–
y
x
1
1
ᎏ
.
Example

Determine the slope of the line joining points A(–3,5) and B(1,–4).
Let (x
1
,y
1
) represent point A and let (x
2
,y
2
) represent point B. This means that x
1
= –3, y
1
= 5, x
2
= 1,
and y
2
= –4. Substituting these values into the formula gives us:
m =
ᎏ
x
y
2
2
–
–
y
x
1

1
ᎏ
m =
ᎏ
1
–
–
4
(
–
–
5
3)
ᎏ
m =
ᎏ
–
4
9
ᎏ
Example
Determine the slope of the line graphed below.
Two points that can be easily determined on the graph are (3,1) and (0,–1). Let (3,1) = (x
1
, y
1
), and
let (0,–1) = (x
2
, y

2
). This means that x
1
= 3, y
1
= 1, x
2
= 0, and y
2
= –1. Substituting these values into
the formula gives us:
y
1
4
3
2
–5
–1
–2
–3
–4
1
5
4
32
–5
–1–2–3–4
x
5
– THEA MATH REVIEW–

146
m =
ᎏ
–
0
1
–
–
3
1
ᎏ
m =
ᎏ
–
–
2
3
ᎏ
=
ᎏ
2
3
ᎏ
Note: If you know the slope and at least one point on a line, you can find the coordinate point of other points
on the line. Simply move the required units determined by the slope. For example, from (8,9), given the slope
ᎏ
7
5
ᎏ
,

move up seven units and to the right five units. Another point on the line, thus, is (13,16).
Determining the Equation of a Line
The equation of a line is given by y = mx + b where:
■
y and x are variables such that every coordinate pair (x,y) is on the line
■
m is the slope of the line
■
b is the y-intercept, the y-value at which the line intersects (or intercepts) the y-axis
In order to determine the equation of a line from a graph, determine the slope and y-intercept and substi-
tute it in the appropriate place in the general form of the equation.
Example
Determine the equation of the line in the graph below.
y
4
2
–2
–4
4
2
–2–4
x
– THEA MATH REVIEW–
147
In order to determine the slope of the line, choose two points that can be easily determined on the
graph. Two easy points are (–1,4) and (1,–4). Let (–1,4) = (x
1
, y
1
), and let (1,–4) = (x

2
, y
2
). This
means that x
1
= –1, y
1
= 4, x
2
= 1, and y
2
= –4. Substituting these values into the formula gives us:
m =
ᎏ
1
–
–
4
(
–
–
4
1)
ᎏ
=
ᎏ
–
2
8

ᎏ
= – 4.
Looking at the graph, we can see that the line crosses the y-axis at the point (0,0). The y-coordinate
of this point is 0. This is the y-intercept.
Substituting these values into the general formula gives us y = –4x + 0, or just y = –4x.
Example
Determine the equation of the line in the graph below.
Two points that can be easily determined on the graph are (–3,2) and (3,6). Let (–3,2) = (x
1
,y
1
), and
let (3,6) = (x
2
,y
2
). Substituting these values into the formula gives us:
m =
ᎏ
3
6
–
–
(–
2
3)
ᎏ
=
ᎏ
4

6
ᎏ
=
ᎏ
2
3
ᎏ
.
We can see from the graph that the line crosses the y-axis at the point (0,4). This means the
y-intercept is 4.
Substituting these values into the general formula gives us y =
ᎏ
2
3
ᎏ
x + 4.
y
4
2
–2
–4
42
–2–4
x
6
–6
–6
6
– THEA MATH REVIEW–
148

Angles
NAMING ANGLES
An angle is a figure composed of two rays or line segments joined at their endpoints. The point at which the rays
or line segments meet is called the vertex of the angle. Angles are usually named by three capital letters, where
the first and last letter are points on the end of the rays, and the middle letter is the vertex.
This angle can either be named either ∠ABC or ∠CBA, but because the vertex of the angle is point B,letter
B must be in the middle.
We can sometimes name an angle by its vertex, as long as there is no ambiguity in the diagram. For exam-
ple, in the angle above, we may call the angle ∠B, because there is only one angle in the diagram that has B as its
vertex.
But, in the following diagram, there are a number of angles which have point B as their vertex, so we must
name each angle in the diagram with three letters.
Angles may also be numbered (not measured) with numbers written between the sides of the angles, on the
interior of the angle, near the vertex.
CLASSIFYING ANGLES
The unit of measure for angles is the degree.
Angles can be classified into the following categories: acute, right, obtuse, and straight.
1
B
C
A
F
D
E
G
B
C
A
– THEA MATH REVIEW–
149

■
An acute angle is an angle that measures between 0 and 90 degrees.
■
A right angle is an angle that measures exactly 90°. A right angle is symbolized by a square at the vertex.
■
An obtuse angle is an angle that measures more than 90°, but less than 180°.
■
A straight angle is an angle that measures 180°. Thus, both of its sides form a line.
Straight Angle
180°
Obtuse Angle
Right
Angle
Symbol
A
cute
Angle
– THEA MATH REVIEW–
150