Sat math Essentials

volume
the number of cubic units inside a three-dimensional figure
F o r m u l a s
The formulas below for area and volume will be provided to you on the SAT. You do not need to memorize them
(although it wouldn’t hurt!). Regardless, be sure you understand them thoroughly.
Circle
Rectangle
Triangle
r
l
w
h
b
A = lw
C = 
2
π
r
A = 
π
r
2
Cylinder
Rectangle
S
olid
h
l
V = 
π
r
2
h
w
r
h
V = lwh
C = 
Circumference
A =
Area
r =
Radiu
s
l =
Length
w =
Width
h =
Height
V =
Volume
b =
Ba
s
e
A
=
1
2
bh
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A n g l e s
An 
angle
is formed by two rays and an endpoint or line segments that meet at a point, called the
vertex
.
Naming Angles
There are three ways to name an angle.
1.
An angle can be named by the vertex when no other angles share the same vertex:
∠
A
.
2.
An angle can be represented by a number or variable written across from the vertex:
∠
1 and 
∠
2.
3.
When more than one angle has the same vertex, three letters are used, with the vertex always being the
middle letter:
∠
1 can be written as 
∠
BAD
or 
∠
DAB,
and 
∠
2 can be written as 
∠
DAC
or 
∠
CAD
.
The Measure of an Angle
The notation m
∠
A
is used when referring to the measure of an angle (in this case, angle 
A
). For example, if
∠
D
measures 100°, then m
∠
D
100°.
1
2
A
C
D
B
vertex
ray #1
ray #2
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Classifying Angles
Angles are classified into four categories: acute, right, obtuse, and straight.
■
An 
acute angle
measures less than 90°.
■
A 
right angle
measures exactly 90°. A right angle is symbolized by a square at the vertex.
■
An 
obtuse angle
measures more than 90° but less then 180°.
■
A 
straight angle
measures exactly 180°. A straight angle forms a line.
S
traight Angle
Obtuse Angle
Right
Angle
Acute
Angle
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Practice Question
Which of the following must be true about the sum of m
∠
A
and m
∠
B
?
a.
It is equal to 180°.
b.
It is less than 180°.
c.
It is greater than 180°.
d.
It is equal to 360°.
e.
It is greater than 360°.
Answer
c.
Both 
∠
A
and 
∠
B
are obtuse, so they are both greater than 90°. Therefore, if 90°
90°
180°, then the
sum of m
∠
A
and m
∠
B
must be greater than 180°.
Complementary Angles
Two angles are 
complementary
if the sum of their measures is 90°.
Supplementary Angles
Two angles are 
supplementary
if the sum of their measures is 180°.
2
1
m
∠
1 + m
∠
2 = 180
S
upplementary
Angle
s
1
2
m
∠
1 + m
∠
2 = 90°
Complementary 
Angles
A
B
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Adjacent
angles have the same vertex, share one side, and do not overlap.
The sum of all adjacent angles around the same vertex is equal to 360°.
Practice Question
Which of the following must be the value of
y
?
a.
38
b.
52
c.
90
d.
142
e.
180
38˚
y
˚
2
1
4
3
m
∠
1 + m
∠
2 + m
∠
3 + m
∠
4 = 360°
1
2
∠
1 and 
∠
2 are adjacent
Adjacent 
Angles
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Answer
b.
The figure shows two complementary angles, which means the sum of the angles equals 90°. If one of
the angles is 38°, then the other angle is (90°
38°). Therefore,
y
°
90°
38°
52°, so 
y
52.
Angles of Intersecting Lines
When two lines intersect,
vertical angles
are formed. In the figure below,
∠
1 and 
∠
3 are vertical angles and 
∠
2
and 
∠
4 are vertical angles.
Vertical angles have equal measures:
■
m
∠
1 
m
∠
3 
■
m
∠
2 
m
∠
4 
Vertical angles are supplementary to adjacent angles. The sum of a vertical angle and its adjacent angle is 180°:
■
m
∠
1 
m
∠
2 
180°
■
m
∠
2 
m
∠
3 
180°
■
m
∠
3 
m
∠
4 
180°
■
m
∠
1 
m
∠
4 
180°
Practice Question
What is the value of
b
in the figure above?
a.
20
b.
30
c.
45
d.
60
e.
120
b
˚
6
a
˚
3
a
˚
2
1
4
3
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Answer
d.
The drawing shows angles formed by intersecting lines. The laws of intersecting lines tell us that 3
a
°
b
° because they are the measures of opposite angles. We also know that 3
a
°
6
a
°
180° because 3
a
°
and 6
a
° are measures of supplementary angles. Therefore, we can solve for 
a
:
3
a
6
a
180
9
a
180
a
20
Because 3
a
°
b
°, we can solve for 
b
by substituting 20 for 
a
:
3
a
b
3(20) 
b
60 
b
Bisecting Angles and Line Segments
A line or segment 
bisects
a line segment when it divides the second segment into two equal parts.
The dotted line 
bisects
segment 
A
B
at point 
C
, so 
A
C
C
B
.
A line 
bisects
an angle when it divides the angle into two equal smaller angles.
According to the figure, ray 
AC
bisects 
∠
A
because it divides the right angle into two 45° angles.
45
45
A
C
A
C
B
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Angles Formed with Parallel Lines
Vertical angles
are the opposite angles formed by the intersection of any two lines. In the figure below,
∠
1 and
∠
3 are vertical angles because they are opposite each other.
∠
2 and 
∠
4 are also vertical angles.
A special case of vertical angles occurs when a transversal line intersects two parallel lines.
The following rules are true when a transversal line intersects two parallel lines.
■
There are four sets of vertical angles:
∠
1 and 
∠
3
∠
2 and 
∠
4
∠
5 and 
∠
7
∠
6 and 
∠
8
■
Four of these vertical angles are obtuse:
∠
1,
∠
3,
∠
5, and 
∠
7
■
Four of these vertical angles are acute:
∠
2,
∠
4,
∠
6, and 
∠
8
■
The obtuse angles are equal:
∠
1 
∠
3 
∠
5 
∠
7
■
The acute angles are equal:
∠
2 
∠
4 
∠
6 
∠
8
■
In this situation, any acute angle added to any obtuse angle is supplementary.
m
∠
1 
m
∠
2 
180°
m
∠
2 
m
∠
3 
180°
m
∠
3 
m
∠
4 
180°
m
∠
1 
m
∠
4 
180°
m
∠
5 
m
∠
6 
180°
m
∠
6 
m
∠
7 
180°
m
∠
7 
m
∠
8 
180°
m
∠
5 
m
∠
8 
180°
1
tran
s
ver
s
al
2
5
6
4
3
7
8
1
2
3
4
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You can use these rules of vertical angles to solve problems.
Example
In the figure below, if
c
|| 
d
, what is the value of
x
?
Because 
c
|| 
d,
you know that the sum of an acute angle and an obtuse angle formed by an intersecting line (line
a
) is equal to 180°.
∠
x
is obtuse and 
∠
(
x
30) is acute, so you can set up the equation 
x
(
x
30) 
180.
Now solve for 
x
:
x
(
x
30) 
180
2
x
30 
180
2
x
30 
30 
180 
30
2
x
210
x
105
Therefore, m
∠
x
105°. The acute angle is equal to 180 
105 
75°.
Practice Question
If
p
|| 
q,
which the following is equal to 80?
a.
a
b.
b
c.
c
d.
d
e.
e
Answer
e.
Because 
p
|| 
q,
the angle with measure 80° and the angle with measure 
e
° are corresponding angles, so
they are equivalent. Therefore 
e
°
80°, and 
e
80.
a˚
110
˚
b˚
e˚
x
y
z
c˚
80
˚
d˚
q
p
x°
(
x 
– 30)
°
b
c
d
a
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Interior and Exterior Angles
Exterior angles
are the angles on the outer sides of two lines intersected by a transversal.
Interior angles
are the
angles on the inner sides of two lines intersected by a transversal.
In the figure above:
∠
1,
∠
2,
∠
7, and 
∠
8 are exterior angles.
∠
3,
∠
4,
∠
5, and 
∠
6 are interior angles.
Tr i a n g l e s
Angles of a Triangle
The measures of the three angles in a triangle always add up to 180°.
Exterior Angles of a Triangle
Triangles have three exterior angles.
∠
a
,
∠
b
, and 
∠
c
are the exterior angles of the triangle below.
■
An exterior angle and interior angle that share the same vertex are supplementary:
a
b
3
1
2
c
3
1
2
m
∠
1 + m
∠
2 + m
∠
3 = 180°
1
tran
s
ver
s
al
2
5
6
4
3
7
8
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m
∠
1
m
∠
a
180°
m
∠
2
m
∠
b
180°
m
∠
3
m
∠
c
180°
■
An exterior angle is equal to the sum of the non-adjacent interior angles:
m
∠
a
m
∠
2 
m
∠
3
m
∠
b
m
∠
1 
m
∠
3
m
∠
c
m
∠
1 
m
∠
2
The sum of the exterior angles of any triangle is 360°.
Practice Question
Based on the figure, which of the following must be true?