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II. Geometry and Measurement
The Praxis II Middle School Content Examination emphasizes your ability to
apply mathematical procedures and algorithms to solve a variety of problems
that span multiple mathematics content areas. Two related content areas are
geometry and measurement. According to the PRAXIS II Study Guide, these
content areas account for 25% of the mathematics component of the Middle
School Content Examination and include the following topics.
Geometry (20%): Examinees should have knowledge of relationships in both two and three
dimensions, as well as draw inferences on the concepts of parallelism, perpendicularity,
congruence and similarity, angle measure, and polygons.
Measurement (5%): Examinees should have knowledge and application of standard units of
both the English and metric systems, nonstandard units, estimation, perimeter, area, volume,
mass, weight, angle measure, time and temperature.
Geometrical shapes and solids are all around us. Look out the window and you may see angles,
triangles, squares, or circles. In ancient times, people built and interacted with a variety of
objects and structures. The shapes and solids that recurred most often were named and their
properties were discovered. In this section, you will explore the properties of these twodimensional shapes.
Topic A: Geometry
A1. Triangles
Triangles come in many shapes and sizes. In nature, we often
see triangles in support structures for buildings and bridges.
Triangles can be classified by their sides or their angles. A
triangle can have no equal sides, two equal sides, or three
equal sides. It can also have three acute angles (angles that are
less than 90°), one right angle, or one obtuse angle (angles that
are greater than 90°). The names of these triangles are shown
in the table below.
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
The sides of a scalene triangle
are all different lengths.
Scalene triangles can be acute,
obtuse, or right triangles.
An isosceles triangle has two
sides that are the same length
and two interior angles that
are the same measure.
Isosceles triangles can be
acute, obtuse, or right
triangles.
All three sides of an
equilateral triangle are the
same length and all interior
angles measure 60°. All
equilateral triangles are acute
triangles.
Obtuse Triangle
Right Triangle
Acute Triangle
A right triangle has one angle
Exactly one of the interior
that is a right angle (as marked
angles of an obtuse triangle is above). In a right triangle the
obtuse (an obtuse angle is over side opposite the right angle is
90 degrees). Obtuse triangles
the longest side and called the
can be scalene or isosceles
hypotenuse. Right triangles
triangles
can be scalene or isosceles
triangles.
All three interior angles of an
acute triangle are acute (less
than 90 degrees). All
equilateral triangles are acute
triangles and some isosceles
and scalene triangles are also
acute triangles.
A Venn Diagram is a graphical organizer that shows the relationships between groups of
things. Using the definitions above, we can create a Venn Diagram to so the relation between
triangles. This diagram shows that all equilateral triangles are acute triangles. No equilateral
triangle can be a scalene triangle. Some acute triangles are scalene triangles and vice versa.
Acute Triangles
Equilateral
Triangles
Scalene Triangles
There is one important property of triangle that deals with the interior angles of a triangle
and one property that deals with the sides of the triangle.
Property 1: The sum of the interior
angles of a triangle sum to 180°.
Property 2: The sum of any two sides of a triangle must
be greater than the third side.
3+4>6
4+6>3
6+3>4
To practice Property 2, please visit the Triangle Constructed Response or Lengths of Sides of
Triangle Learning Objects.
The Pythagorean Theorem, dealing with the relationships in right triangles, is one of the
most famous theorems in geometry. This theorem describes the relationship between the lengths
of legs of a right triangle and the hypotenuse.
Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs
is equal to the square of the hypotenuse. When the lengths of the legs are a and b, and the
length of the hypotenuse is c, then a2 + b2 = c2.
In the following learning objects, you will apply Pythagorean’s Theorem to solve problems.
• Cross Sectional Area
• Coordinate Geometry
A2. Quadrilaterals
The word quadrilateral is derived from two Latin words quadri- which means four
and later- which means sides. Therefore, a quadrilateral is a polygon with four sides.
There are many different quadrilaterals that have important geometric properties as
described in the table below.
A trapezoid is a quadrilateral that has exactly one pair of parallel sides. An isosceles trapezoid
is a trapezoid (in yellow) in which the non-parallel sides are congruent (same length).
A kite is a quadrilateral that has two pair of adjacent sides that are congruent (same length).
A parallelogram is a quadrilateral in which each pair of opposite sides are parallel. This means
that opposite sides are also congruent (or the same length). In a parallelogram, the opposite
angles are congruent (or the same measure) and the adjacent angles are called supplementary
angles because they add to 180 degrees.
A rectangle is a parallelogram that has four right angles. Since a rectangle is a parallelogram it
must have opposite sides that are parallel and congruent.
A rhombus is a parallelogram that has four congruent sides. Since a rhombus is a parallelogram
it must have opposite sides that are parallel and congruent.
A square is a rectangle that has four congruent sides. A square is a rhombus that has four right
angles. A square is equilateral (equal side lengths) and equiangular (equal angle measures).
The Venn Diagram below shows the relationship between these special quadrilaterals.
Rectangles
Trapezoids
Squares
Rhombuses
Parallelograms
Below are seven statements about quadrilaterals. Can you identify which ones are always true,
sometimes true, or never true?
1. A trapezoid is a parallelogram.
2. A square is always a rectangle.
3. A rectangle has four right angles.
4. A rhombus always has four equal sides.
5. A parallelogram has one set of opposite sides.
6. A parallelogram is never a square.
7. A rhombus is never a square.
To visualize other geometric relationships, visit the Venn Diagram Learning Object.
There is one important property of quadrilaterals that deals with the interior angles. The
sum of the interior angles in any quadrilateral adds to 360 degrees.
Other properties of quadrilaterals can be
summarized in the following table. The last two
rows provide facts about the diagonals of
quadrilaterals. The diagonals are line segments
drawn from one vertex to the opposite vertex as
shown in the parallelogram to the right.
Properties of
Quadrilaterals
Rectangle
Square
Never
Never
Never
Isosceles
Trapezoid
Always
Always
Never
Always
Always
Never
Always
Always
Parallelogram Rhombus
Never
Exactly 1 pair of
parallel sides
2 pairs of parallel sides Always
Exactly 1 pair of
congruent sides
Always
2 pairs of congruent
sides
All sides congruent
Always
Trapezoid
Kite
Always
Never
Never
Always
Never
Never
Never
Always
Never
Never
Always
Always
Never
Never
Never
Opposite angles
congruent
Adjacent angles are
supplementary (add to
180 degrees)
Perpendicular
diagonals
Congruent diagonals
Always
Always
Always
Always
Never
Never
Always
Always
Always
Always
Always
Never
Never
Always
Always
Never
Never
Always
Always
Always
Always
Always
Never
A3. Other Polygons
A polygon is a many sided figure that has an equal number of sides as
angles. Triangles and quadrilaterals are two examples of polygons.
Other polygons are name according to the number of sides (or number
of angles) that they have. The picture to the right is a pentagon
because it has five sides. A polygon that has congruent angles and
congruent sides is called a regular polygon. A regular polygon is
equiangular and equilateral.
Polygon Names
Number of Sides
3
4
5
6
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Number of Sides
7
8
9
10
Name
Heptagon
Octagon
Nonagon
Decagon
In a triangle the sum of the interior angles is 180°. In a rectangle the sum of the four interior
angles is 360°. This fact can be extended to all quadrilaterals. The process below to verify this
property can be extended to all polygons.
The sum of the interior angles of a quadrilateral is equal to 360°.
In this quadrilateral we can draw a diagonal from one
vertex to the opposite vertex creating two triangles. The
sum of the interior angles in each of the triangles is 180°.
This makes the sum of the interior angles of a
quadrilateral equal two 180° + 180° = 360°
The sum of the interior angles of a quadrilateral is equal to 540°.
In this pentagon we can draw two diagonals from one
vertex to the opposite vertices creating three triangles.
The sum of the interior angles in each of the triangles is
180°. This makes the sum of the interior angles of a
pentagon equal two 180° + 180° + 180° = 540°
In the learning object, Polygon Sum Conjecture, you will examine how to find the sum of the
interior angles for any polygon.
A4. Symmetry
Symmetry adds beauty and balance to natural forms and
architectural design. The Wisconsin Model Academic
Standards for Mathematics Glossary
(http://dpi.wi.gov/standards/mathglos.html)of terms
defines symmetry as follows. A figure has symmetry if
it has parts that correspond with each other in terms of
size, form, and arrangement. There are two common
types of symmetry shown in the pictures to the left. The
first column represents reflection (line) symmetry and
the second column represents rotational symmetry.
A figure with reflection symmetry has two halves which match each other perfectly if the figure
is folded along its line. This type of symmetry is also called line or mirror symmetry. A plane
figure may have more than one line of symmetry as shown below.
A figure has rotation symmetry or turn symmetry if the figure comes back to itself when rotated
through a certain angle between 0° and 360°. Many shapes have rotational symmetry. For
example, a rectangle can be turned 180° and it will come back to itself. An equilateral triangle
has two rotation symmetries at 120° and 240°. A figure with a 90° rotation (like the leafs and
flowers below) will also have 180° and 270°.
A5. Transformations
Two polygons are congruent if they have the same shape and the same size. In geometry, two
polygons are congruent if their corresponding sides are equal in length and their corresponding
angles are equal in measure. As an example, the two triangles below are congruent.
ΔABC ≅ ΔDEF
If you ask a child to determine if two polygons are congruent, they may place one shape over the
other. Consider the types of moves a child would have to make to “see” that the triangles below
are congruent.
The set of rigid transformations consists of the various ways to move a geometric figure around
while preserving the distance between the points in the figure. Rigid transformations always
produce congruent figures. In the learning object Reflection, you will explore these geometrical
transformations.
A6. Coordinate Geometry
Coordinate Geometry
The coordinate plane is a basic concept for coordinate
geometry. The coordinate plane is also called the xyplane and it is useful for plotting points, lines, and figure.
In the coordinate plane to the right, point L is represented
by the coordinates (–3, 1.5) because it is positioned on –3
along the x-axis and on 1.5 along the y-axis. Similarly,
you can figure out why the points M = (2, 1.5) and N = (–
2, –3).
In coordinate geometry, we often use the xy plane
to find the distance between two points. As an example,
what is the distance between points M and N. To do this we can construct a right triangle where
the line segment joining points M and N serves has the hypotenuse of this triangle.
The distance between the two points is the length of
the hypotenuse which can be found using
Pythagorean’s Theorem.
c2 = 42 + 4.52
c2 = 36.25
c = 36.25 ≈ 6.02 units
For more information, visit the following learning object.
• Coordinate Geometry
• Points on a Line
Topic B Measurement
B1.Standard Units of Measurement
What does it mean to measure something? Measurement is the assignment of a numerical value
to an attribute of an object, such as the length of a pencil, the area of a playground, the weight of
a rock, or the temperature of a hot cup of coffee. The units that we use to measure are most often
standard units, which means that they are universally available and are the same size for all who
use them. Sometimes we measure using nonstandard units, which means that we are using units
that we have invented. For example, we could measure the length of a room in arm spans instead
of meters.
Two standard measurement systems have been developed and universally agreed upon. These
systems of measurement are called the English System and the Metric System.
The most frequently used units of measurement in the English System are provided below.
Weight/Mass: Pound
Length: Foot
Capacity/Volume:
Temperature:
Gallon
Fahrenheit
16 ounces = 1 pound
1 yard = 3 feet
1 mile = 5,280 feet
1 pint = 2 cups
12 inches = 1 foot
1 quart = 2 pints
1 gallon = 4 quarts
The Metric System resembles our decimal system and is based on powers of ten. Conversions
are made by moving the decimal place to the left or right. The standard unit of length in the
metric system is the meter. Other units of length such as the kilometer and the centimeter can be
found by multiplying by the appropriate power of 10 as described in the table below.
kilohectodecaunit
decicenti milli-
1000
100
10
thousands
hundreds
tens
1000 meters = 1 kilometer (km)
100 meters = 1 hectometer
10 meters = 1 decameter
0.1
0.01
0.001
tenths
hundredths
thousandths
1 meter = 10 decimeters
1 meter = 100 centimeters (cm)
1 meter = 1000 millimeters (mm)
In the metric system, the basic unit for weight/mass is the gram. The basic unit for
capacity/volume is liters. The basic unit for temperature is Celsius.
Take the short Metric Mayhem Quiz to test your ability to compare metric measurements or
visit the Units of Conversion learning object.
There are standard tables to convert from the English system to the metric system. Here are
some examples.
•
This paperclip has a length of 1 inch and weighs 1 gram.
. We could also
say that this paperclip has a length of about 2.54 centimeters and weighs about 0.353
ounces.
• The ruler below is 1 foot long. Look closely because you will also see that it is about 30
centimeters long. Actually 1 foot = 30.48 centimeters = 0.3048 meters.
•
•
A gallon of water
is equivalent to 3.785 liters.
The optimal temperature for a bath is 38 degrees Celsius. This is equivalent
to 100 degrees Fahrenheit. The typical temperature conversion formula is
given by F = 9/5C + 32.
Unit Analysis
Unit analysis or dimensional analysis is a procedure fro converting from one unit of
measurement to another unit of measurement. It uses proportional reasoning to transform form
one unit to the next without having to know all the standard conversion factors. For example,
how many meters are in the 400-yard dash? Since we do not know how many meters in a yard,
we will use the fact that 1 foot = 0.3048 meters to solve this problem.
400 yards ×
3 feet 0.3048 meters
×
= (400 ⋅ 3 ⋅ 0.3048) meters = 365.76 meters
1 yard
1 foot
If Mai finishes the 400-yard dash in 8 minutes, then how fast did she run in miles per hour? To
answer this question we will use the fact that 5,280 feet = 1,760 yards = 1mile.
400 yards
1 mile
60 min
×
×
= (400 × 60 ) ÷ (4 × 1760) miles per hour ≈ 3.41 miles per hour
4 min
17760 yards 1 hour
To learn more about unit analysis, visit the following learning object.
• Units of Measure
• Unit Analysis
• Proportional Reasoning
B2. Area and Perimeter
Area and Perimeter
The area of a shape is typically defined as the number of square units that are required to
completely cover that shape. In the picture below, the area of the rectangle is 35 square units.
Common units of measurement for area are square centimeters (cm2) or square feet (ft2).
The perimeter is a measure of length so it is given the units centimeters or feet. The perimeter
can be translated as the “around measure” or the length of the closed curve. For a polygon, the
perimeter is the sum of the lengths of all its sides. For a circle, the perimeter is often referred to
as the circumference.
The most common two dimensional figures for which areas and perimeters must be found
are triangles, parallelograms, rectangles, squares, trapezoids, and circles. This section will
develop a formula for finding the area of these polygons.
Rectangles
Squares
Area = s2
Perimeter = 4s
Height = h
Base = b
Area = bh
Perimeter 2b + 2h
Parallelogram
Area = (area of rectangle) = bh
Perimeter: Add up lenghts of 4 sides.
Circle
Side = s
Triangle
Area = ½(area of parallelogram) = 1/2bh
Perimeter: Add up the lengths of the 3 sides
Trapezoid
Area = πr2
Circumference = 2πr = πd
Area = (1/2 area of parallelogram) = ½(a + b)h
Perimeter: Add up the lengths of the 4 side
Visit the following learning objects to practice your skills using these formulas.
• Area Learning Object
• Dimensions
• Cross Section Area
To learn more about circles, visit the Properties of Circles Leaning Object.
B3. Similar Figures
In geometry, two polygons are similar if they have the same shape (not necessarily the
same size). The picture below gives examples of similar polygons.
Triangles are similar if their corresponding (matching) angles are equal and the ratio of their
corresponding sides are in proportion.
Corresponding Sides are in
ΔABC ~ ΔDEF
Proportion
6
8 10 1
=
=
=
12 16 20 2
Quadrilaterals are similar if their corresponding (matching) angles are equal and the ratio of their
corresponding sides are in proportion.
Corresponding Sides are in
Proportion
2 3 1
= =
6 9 3
Similar Triangles: Corresponding sides of the similar triangles below are proportional since
6
8 10 1
= =
= . What is the ratio of the triangles perimeters? The ratio of the perimeters is
12 16 20 2
also proportional to the ratio of
corresponding sides,
24 1
= . In other words, if two
48 2
polygons are similar, their
corresponding sides, heights,
diagonals, and perimeters are all
in the same ratio.
Perimeter =
12 + 16 + 20
= 48 units
Perimeter =
6 + 8 + 10 =
24 units
Similar Rectangles: Corresponding sides of the similar rectangles below are proportional since
2
6 1 ⎛1⎞
2 3 1
= = ⎜ ⎟ . In
= = . What is the ratio of the areas? The ratio of the areas is given by
54 9 ⎝ 3 ⎠
6 9 3
other words, if two polygons are similar, the ratio of their areas is equal to the square of the
ratio of their corresponding sides.
Area: 9 x 6 = 54
square units
Area:
3x2=6
square units
To work with similar polygons, visit the Ratio of Inscribed Square learning objects.
B4. Surface Area and Volume
Three-dimensional figures or space figures can also be found right
outside your window. Many every day objects, such as boxes, cans,
wedges, and spheres, represent special geometrical solids. If a solid is
made up of polygonal faces, then we call this solid a polyhedron. For
example a box and wedge are polyhedra. The correct mathematical term
to describe a polyhedron is to name it according to the number of faces.
A box has six faces so we refer it to as a hexahedron. The wedge is a
pentahedron with five faces. Cones, cylinders, and spheres are special
types of solids that are not examples of polyhedra.
The cereal box and the cheese wedge are common
prisms. A prism has two bases that are congruent
polygonal regions that lie in parallel planes. The
lateral faces that join the base are all parallelograms. A
box is also referred to as a rectangular prism and it has
8 vertices and 12 edges. The wedge is referred to as a
triangular prism with 6 vertices and nine edges.
Other special polyhedrons can also be found in nature and
architecture. The Great Pyramid of Giza represents a pentahedron
since it has a square base and four triangular faces. In geometry, a
pyramid has triangular faces which meet a common point called the
apex. The base of a pyramid can be any polygonal region which is
used to name the pyramid. Polyhedra can be constructed from twodimensional shapes called nets that can be folded into the
appropriate form. Can you name which net belongs to the given
polyhedra?
How much will a container cost to manufacture? How much paint do you need to paint a
bedroom? To answer these questions you must find the surface area. The total surface area is
the sum of the area of all of its surfaces. A surface area in square units indicates how many
squares it would take to wrap the outside of the space figure. The surface area is also the
combined two-dimensional area of a geometrical net. There are many formulas to find surface
are as shown in the table below.
Solid
Net
Rectangular Prism
Formula
2 x area of rectangular base +
4 x area of rectangular sides.
Square Pyramid
Area of square base + 4 x
area of triangular sides.
2 x are of circular bases +
area of rectangular side
Cylinder
Volume is the amount of space occupied by a three-dimensional figure.
Volume is typically measured in cubic units, but may also be described as
gallons, fluid ounces, or even the number of people required to “fill” a
phone booth. Whereas, surface area is the total areal of the faces of the
solid, volume is the capacity of the solid. In other words, surface area can
be thought of as wrapping paper, volume can be considered as the filling.
As with surface area, there are many known formulas for calculating the
volume of prisms, cylinders, pyramids, and other space figures.
Solid
Rectangular Prism
Triangular Prism
Cylinder
Examples
Formula
Area of the
rectangular Base B = l
xw
Volume V = l x w x h
= Bh
where B is the area of
the base.
Area of triangular
base B = ½bh
Volume V = Bh
where B = area of the
base.
Area of the circular
Base B = πr2.
Volume V = πr2h
= Bh
where B is that area of
the base.
Generalization
Prisms and
Cylinders
V = Bh
where B = area of
the base
Square Pyramid
Area of square Base B
= b2
2
Volume V = 1/3b h
= 1/3Bh
where B is the area of
the base.
Pyramids
Volume V =
1/3Bh
where B is the
area of the base.