Download The slope of a line can be found if you know the coordinates of any

– THEA MATH REVIEW –
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.)
y2 – y1
The formula for the slope of a line (or line segment) containing points (x1, y1) and (x2, y2): m = x –x .
2
1
Example
Determine the slope of the line joining points A(–3,5) and B(1,–4).
Let (x1,y1) represent point A and let (x2,y2) represent point B. This means that x1 = –3, y1 = 5, x2 = 1,
and y2 = –4. Substituting these values into the formula gives us:
y2 – y1
m=
x –
x
2
1
–4 – 5
m=
1 – (–3)
m = –49
Example
Determine the slope of the line graphed below.
y
5
4
3
2
1
x
–5
–4
–3
–2
1
–1
2
3
4
5
–1
–2
–3
–4
–5
Two points that can be easily determined on the graph are (3,1) and (0,–1). Let (3,1) = (x1, y1), and
let (0,–1) = (x2, y2). This means that x1 = 3, y1 = 1, x2 = 0, and y2 = –1. Substituting these values into
the formula gives us:
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–1 – 1
m=
0–3
2
m = ––23 = 3
Note: If you know the slope and at least one point on a line, you can find the coordinate point of other points
7
on the line. Simply move the required units determined by the slope. For example, from (8,9), given the slope 5,
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 substitute 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
x
–4
2
–2
–2
–4
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– THEA MATH REVIEW –
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) = (x1, y1), and let (1,–4) = (x2, y2). This
means that x1 = –1, y1 = 4, x2 = 1, and y2 = –4. Substituting these values into the formula gives us:
–4–4
–8
m=
1 – ( – 1) = 2 = – 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.
y
6
4
2
x
–6
–4
2
–2
4
6
–2
–4
–6
Two points that can be easily determined on the graph are (–3,2) and (3,6). Let (–3,2) = (x1,y1), and
let (3,6) = (x2,y2). Substituting these values into the formula gives us:
4
2
6–2
m=
3 – (–
3) = 6 = 3 .
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.
2
Substituting these values into the general formula gives us y = 3x + 4.
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Angles
N AMING A NGLES
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.
A
B
C
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 example, 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.
C
D
A
B
G
E
F
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.
1
C LASSIFYING A NGLES
The unit of measure for angles is the degree.
Angles can be classified into the following categories: acute, right, obtuse, and straight.
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■
An acute angle is an angle that measures between 0 and 90 degrees.
Acute
Angle
■
A right angle is an angle that measures exactly 90°. A right angle is symbolized by a square at the vertex.
Right
Angle
Symbol
■
An obtuse angle is an angle that measures more than 90°, but less than 180°.
Obtuse Angle
■
A straight angle is an angle that measures 180°. Thus, both of its sides form a line.
Straight Angle
180°
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S PECIAL A NGLE PAIRS
■
Adjacent angles are two angles that share a common vertex and a common side. There is no numerical
relationship between the measures of the angles.
1
2
2
1
Adjacent angles ∠1 and ∠2
■
■
A linear pair is a pair of adjacent angles whose measures add to 180°.
Supplementary angles are any two angles whose sum is 180°. A linear pair is a special case of supplementary angles. A linear pair is always supplementary, but supplementary angles do not have to form a linear
pair.
70˚
110˚
70˚
110˚
Linear pair (also supplementary)
■
Non-adjacent angles ∠1 and ∠2
Supplementary angles (but not a linear pair)
Complementary angles are two angles whose sum measures 90 degrees. Complementary angles may or
may not be adjacent.
40˚
40˚
50˚
Adjacent complementary angles
50˚
Non-adjacent complementary angles
Example
Two complementary angles have measures 2x° and 3x + 20°. What are the measures of the angles?
Since the angles are complementary, their sum is 90°. We can set up an equation to let us solve for x:
2x + 3x + 20 = 90
5x + 20 = 90
5x = 70
x = 14
Substituting x = 14 into the measures of the two angles, we get 2(14) = 28° and 3(14) + 20 = 62°. We
can check our answers by observing that 28 + 62 = 90, so the angles are indeed complementary.
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Example
One angle is 40 more than 6 times its supplement. What are the measures of the angles?
Let x = one angle.
Let 6x + 40 = its supplement.
Since the angles are supplementary, their sum is 180°. We can set up an equation to let us solve for x:
x + 6x + 40 = 180
7x + 40 = 180
7x = 140
x = 20
Substituting x = 20 into the measures of the two angles, we see that one of the angles is 20° and its
supplement is 6(20) + 40 = 160°. We can check our answers by observing that 20 + 160 = 180, proving that the angles are supplementary.
Note: A good way to remember the difference between supplementary and complementary angles is that the
letter c comes before s in the alphabet; likewise “90” comes before “180” numerically.
A NGLES
OF
I NTERSECTING L INES
Important mathematical relationships between angles are formed when lines intersect. When two lines intersect,
four smaller angles are formed.
4
3
1
2
Any two adjacent angles formed when two lines intersect form a linear pair, therefore they are supplementary. In this diagram, ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, and ∠4 and ∠1 are all examples of linear pairs.
Also, the angles that are opposite each other are called vertical angles. Vertical angles are angles who share
a vertex and whose sides are two pairs of opposite rays. Vertical angles are congruent. In this diagram, ∠1 and ∠3
are vertical angles, so ∠1 ≅ ∠3; ∠2 and ∠4 are congruent vertical angles as well.
Note: Vertical angles is a name given to a special angle pair. Try not to confuse this with right angle or perpendicular angles, which often have vertical components.
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Example
Determine the value of y in the diagram below:
3y + 5
5y
The angles marked 3y + 5 and 5y are vertical angles, so they are congruent and their measures are
equal. We can set up and solve the following equation for y:
3y + 5 = 5y
5 = 2y
2.5 = y
Replacing y with the value 2.5 gives us the 3(2.5) + 5 = 12.5 and 5(2.5) = 12.5. This proves that the
two vertical angles are congruent, with each measuring 12.5°.
PARALLEL L INES
AND
T RANSVERSALS
Important mathematical relationships are formed when two parallel lines are intersected by a third, non-parallel
line called a transversal.
l
2
1
4
m
3
6
5
8
7
n
In the diagram above, parallel lines l and m are intersected by transversal n. Supplementary angle pairs and
vertical angle pairs are formed in this diagram, too.
Supplementary Angle Pairs
Vertical Angle Pairs
∠1 and ∠2
∠2 and ∠4
∠1 and ∠4
∠4 and ∠3
∠3 and ∠1
∠2 and ∠3
∠5 and ∠6
∠6 and ∠8
∠5 and ∠8
∠8 and ∠7
∠7 and ∠5
∠6 and ∠7
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Other congruent angle pairs are formed:
■
■
Alternate interior angles are angles on the interior of the parallel lines, on alternate sides of the transversal: ∠3 and ∠6; ∠4 and ∠5.
Corresponding angles are angles on corresponding sides of the parallel lines, on corresponding sides of
the transversal: ∠1 and ∠5; ∠2 and ∠6; ∠3 and ∠7; ∠4 and ∠8.
Example
In the diagram below, line l is parallel to line m. Determine the value of x.
4x + 10
l
m
8x – 25
n
The two angles labeled are corresponding angle pairs, because they are located on top of the parallel
lines and on the same side of the transversal (same relative location). This means that they are congruent, and we can determine the value of x by solving the equation:
4x + 10 = 8x – 25
10 = 4x – 25
35 = 4x
8.75 = x
We can check our answer by replacing the value 8.75 in for x in the expressions 4x + 10 and 8x – 25:
4(8.75) + 10 = 8(8.75) – 25
45 = 45
Note: If the diagram showed the two angles were a vertical angle pair or alternate interior angle pair, the problem would be solved in the same way.
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– THEA MATH REVIEW –
Area, Circumference, and Volume Formulas
Here are the basic formulas for finding area, circumference, and volume. They will be discussed in detail in the
following sections.
Rectangle
Circle
Triangle
r
w
h
l
C = 2πr
A = πr 2
b
A = 1_2 bh
A = lw
Rectangular
Solid
Cylinder
r
h
h
w
l
V = πr 2 h
C
A
r
l
=
=
=
=
V = lwh
w
h
v
b
Circumference
Area
Radius
Length
=
=
=
=
Width
Height
Volume
Base
Triangles
The sum of the measures of the three angles in a triangle always equals 180 degrees.
b
a
c
a + b + c = 180°
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