Math Review
Geometry and Measurement
Geometric Notation
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You will need to be able to recognize and use geometric notation for points
and lines, line segments, rays, angles and their measures, and lengths.
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mIIp (m is parallel to p)
PE with a line with two arrows on each side means a line containing points P and
E
PE with just a line segment over it means a line with endpoints P and E
PE means the length of the line segment P to E
PE with an arrow to the right only means the ray starting at P and extending
infinitely in the direction E
EP with an arrow above it to the right only means a ray starting at E and
extending infinitely in the direction of P
<DOC means the angle formed by rays OD and OC
m<DOC the measure of angle DOC
(little triangle)OQC means the triangle with vertices O,C, and Q
BPMO means the quadrilateral with vertices B,P,M, and O
BP with a line above it (upside down T) PM with a line above it means the
relation that line segment BP is perpendicular to line segment BM
Points and Lines
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There is a unique line that contains any two distinct
points. Therefore a line l is the only line that contains
both point A and point B.
The midpoint of a line segment is the point that divides it
into two segments of equal lengths.
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Also understand how to add the lengths of line segments
along the same line.
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If M is the midpoint of line segment AB, then you know AM=MB.
If line segment PQ=2 and line segment QR=3, then PR=5
You may also be given the lengths of line segments
along a common line and be asked that requires you to
find the order of points along the line.
Points and Lines
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Example
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Points E,F, and G all lie on line m, with E to
the left of F. EF=10, FG=8, and EG>FG. What
is EG?
 Draw
 10-8
the line
F
E
G
______________
8
10
= 2, so EG is 2, but then it’s not greater than
FG, so try putting G on the other side and add the
two line segments together. 10+8=18, so EG
should be 18 (bigger than FG)
Angles in the Plane
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Vertical Angles
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Supplementary Angles
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Two opposite angles formed by intersecting lines. They have the same
measure.
Two angles whose measures have a sum of 180 degrees
Parallel Lines
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When a line intersects a pair of parallel lines, the eight angles formed
are related in several ways.
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The measures of corresponding angles are equal.
Several pairs of angles are straight and will add up to 180 degrees
Alternate interior angles have equal measures
Right Angle has a measure of 90 degrees
Perpendicular Lines occur when two lines intersect and one of the
four angles is a right angle.
Complementary Angles are two angles whose measures have a
sum of 90 degrees.
Triangles
The sum of the measures of the angles in
any triangle is 180 degrees.
 Equilateral Triangles
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The 3 sides are equal in length. The 3 angles
are also equal and they each measure 60
degrees.
Isosceles Triangles
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Has 2 sides of equal length. The angles
opposite the equal sides are also equal.
Triangles
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Right Triangles
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A triangle with a right angle. The other two angles are
complementary angles.
Pythagorean Theorem
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a^2 + b^2 = c^2
If you know the lengths of any 2 sides, you can use the Pyth
Theorem to find the length of the third side.
30 – 60 – 90 degrees
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The lengths of the sides are in the ratio of 1:(square
root of 3):2
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Short length=x, Long length=x(square root of 3),
Hypotenuse=2x
Triangle
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45-45-90 degrees Triangle
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The lengths of the sides are in the ratio of
1:1:(square root of 2). Two short sides=x and
hypotenuse=x(square root of 2)
3-4-5 Triangles
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The sides are in the ratio of 3:4:5. Short
leg=2x, Long leg=4x, and Hypotenuse=5x
Triangle
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Congruent Triangles
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Two triangles that have the same size and shape.
Each side has the same length of the corresponding side of the
other triangle.
Each angle is equal to its corresponding angle.
Similar Triangles
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Have the same shape. Each corresponding pair of angles has
the same measure.
2 triangles are similar if:
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2 pairs of corresponding angles have each have the same measure
One pair of corresponding angles has the same measure, and the
pairs of corresponding sides that form those angles have lengths
that are in the same ratio.
Triangles
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The Triangle Inequality
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It states that the sum of the lengths of any 2
sides of a triangle is greater than the length of
the 3rd side.
Quadrilaterals
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Parallelograms
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Rectangles
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The opposite angles are of equal measure
and the opposite sides are of equal length.
All angles are right angles.
Squares
All angles are right angles, and all sides are
equal.
 The diagonal will make two 45-45-90 degree
triangles
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Areas and Perimeters
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Areas of Rectangles and Squares
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The formula for the area of any rectangle is:
 Area=length*width
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Area of a square can be written as:
 Area=s^2
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(where s is the length of the side)
Perimeters of Rectangles and Squares
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Perimeter of a polygon is the sum of the
lengths of its sides
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Rectangle: Perimeter=2(length+width)=2(l+w)
 For Square: Perimeter=4(length of any side)=4s
Areas and Perimeters
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Area of Triangles
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A=(1/2)bh
 b=base
 h=height
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Area of Parallelograms
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To find the area of a parallelogram, drop a
perpendicular line down each side to make 2
separate right angles (will make a rectangle).
 Area=length*height
be the width)
(on the “new” rectangle it will
Other Polygons
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Regular Polygon
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Polygon whose sides all have the same length and whose
angles all have the same measure.
Angles in a Polygon
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You an figure out the total number of degrees in the interior
angles of most polygons by dividing the polygon into triangles
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From any vertex, divide the polygon into as many nonoverlapping
triangles as possible. Use only straight lines. Make sure that all the
space inside the polygon is divided into triangles.
Count the triangles.
There is a total of 180 degrees in the angles of each triangle, so
multiply the number of triangles by 180. The product will be the sum
of the angles in the polygon.
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3 sides = 180 degrees
4 sides = 360 degrees
5 sides = 540 degrees
6 sides = 720 degrees
n sides = 180(n-2)
Circles
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Diameter
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Radius
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A line segment that passes through the center and has its
endpoints on the circle. All diameters of the same circle have
equal lengths.
A line segment extending from the center of the circle to a point
on the circle. All radii of the same circle have equal lengths.
Arc
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A part of the circle. Two points on a circle form an arc. It can be
measured in degrees or in units of length.
If you form an angle by drawing radii from the ends of the arc to
the center of the circle, the number of degrees in the arc equals
the number of degrees in the angle formed by the two radii at the
center of the circle, called the Central Angle.
Circles
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Tangent to a Circle
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A line that intersects the circle at exactly one point. It
is always perpendicular to the radius that contains
that one point of the line that touches the circle.
Circumference
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The distance around the circle. It is equal to pie times
the diameter, d (or pie times the radius r)
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Circumference=pie(d)
Circumference=2(pie)(r)
Area
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Is equal to pie times the square of the radius
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Area=(pie)r^2
Solid Geometry
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Solid Figures and Volumes
You will be given the formulas involving solids
but you may be asked questions that require
logic and apply the knowledge in new ways.
 Cubes and Rectangular Solids
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 Volume
for cube=s^3 (s=side)
 Volume for rectangular solid=lwh
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Prisms and Cylinders
 Volume
of prism=height*(area of base)
 Volume of cylinder=(pie)r^2(height)
Solid Geometry
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Spheres, Cones, and Pyramids
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Sphere
 Solid
analogue of a circle. All radii of a sphere are
equal.
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Cone
 Circular
Cone has a circular base which is
connected by a curved to its vertex.
 Right Circular Cone has a line from the vertex of
the circular cone to the center of its base that is
perpendicular to the base.
Solid Geometry
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Surface Area
If you make cuts in a solid that allow you to
open it up to form a plane figure, the result is
called a net of the solid. A solid may have
many different nets.
 You can sometimes find the surface area of a
solid by looking at the nets.
 The sum of the areas of each surface.
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Geometric Perception
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You may be asked to visualize a plane
figure or solid from different views or
orientations.
Coordinate Geometry
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Slopes, Parallel Lines, and Perpendicular
Lines
Some of the geometry questions on the test
will involve points, lines, and figures in the
coordinate plane.
 Slopes
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2
lines are perpendicular when the product of their
slopes is -1.
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The Midpoint Formula
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The average of the x’s and the y’s.
Coordinate Geometry
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The Distance Formula
In the coordinate plane, the Pythagorean
Theorem can be used to find the distance
between any 2 points.
 In general, if two points (x,y) and (m,n) are
two points in the coordinate plane, the
distance between them is given by the
formula
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 D=square
root of [(x-m)^2+(y-n)^2]
Transformations
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Translation
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Rotating
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Turning it around a point (the center of rotation).
Reflection
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Moves a shape without any rotation or reflection.
To produce its mirror image with respect to a line (the line of
reflection). If you reflect a figure twice across the same line, you
get back the original figure.
Symmetry
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If you put a line down the middle of the figure and the halves are
reflections of each other, then the figure is symmetrical. The line
is called the line of symmetry.
There may be more than one line of symmetry.
Another type of symmetry is around a point (the point of
symmetry). If you rotate the figure around a point 180 degrees
and it is the same figure, then it is symmetrical in that way also.