Geometry Study Guide for McGraw-Hill's ASVAB (page 5)
Practice problems for this study guide can be found at:
Geometry On The ASVAB Mathematics Knowledge Test
Along with problems in algebra and probability, the ASVAB Mathematics Knowledge test also includes problems in geometry. To do well on this portion of the test, you'll need to know the basic geometry concepts taught in high school math courses. Topics tested include classifying angles, identifying different kinds of triangles and parallelograms, calculating perimeter and area, finding the circumference and area of circles, identifying different kinds of solid figures, and solving geometry word problems.
It is important that you do well on the Mathematics Knowledge test because it is one of the four ASVAB tests that are used to calculate the AFQT—your military entrance score. That's why it pays to spend time reviewing topics in algebra, probability, and geometry and tackling plenty of sample ASVAB Mathematics Knowledge questions.
The following pages offer a quick but important overview of the basic geometry you need to know to score well on the ASVAB. Make sure that you carefully review and test yourself on every topic covered in this section. Also make sure that you learn how to use all of the problem-solving methods presented in the examples.
Points, Lines, and Angles
To work with geometry, you need to understand points, lines, and angles.
- A point is an exact location in space. It is represented by a dot and a capital letter.
- A line is a set of points that form a straight path extending in either direction without end. A line that includes points B and D is represented as follows: BD.
- A ray is a part of a line that has one endpoint and continues without end in the opposite direction. A ray that ends at point A and includes point B is represented as follows: .
- A line segment is a part of a ray or a line that connects two points. A line connecting points A and B is represented as follows: .
An angle is a figure formed by two rays that have the same endpoint. That endpoint is called the vertex (plural: vertices) of the angle. An example is shown below.
In this example, rays and have the same endpoint, which is point B. So point B is the vertex of the angle. The two line segments are called the sides of the angle. The symbol is used to indicate an angle.
An angle is labeled or identified in several different ways:
- By the vertex: B
- By the letters of the three points that form it: ABC or CBA. (The vertex is always the middle of the three letters.)
The measure of the size of an angle is expressed in degrees (°).
Classifying Angles There are three types of angles that you should know for the ASVAB test. They are right angles, acute angles, and obtuse angles.
Right Angles A right angle measures exactly 90°. Right angles are found in squares, rectangles, and certain triangles. ABC is a right angle.
The angles below are both right angles.
Acute Angles An angle that measures less than 90° is called an acute angle. STU is an acute angle.
The angles below are all acute angles.
Obtuse Angles An angle with a measure that is greater than 90° but less than 180° is called an obtuse angle. MNO is an obtuse angle.
The angles below are all obtuse angles.
Straight Angles A straight angle is one that measures exactly 180°. This kind of angle forms a straight line. EFG is a straight angle.
Classifying Pairs of Lines
Intersecting Lines Intersecting lines are lines that meet or cross each other.
Line DF intersects line GH at point E.
Parallel Lines Parallel lines are lines in a plane that never intersect.
Line MN is parallel to line OP. In symbols, MN || OP.
Perpendicular Lines Perpendicular lines intersect to form right angles.
Line ST is perpendicular to line UV. In symbols, ST UV.
Classifying Pairs of Angles
Adjacent Angles Adjacent angles have the same vertex and share one side. ABC and CBD are adjacent angles.
Complementary Angles Two adjacent angles whose measures total 90° are called complementary angles. MNO and ONP are complementary. Their measures total exactly 90°.
HINT Figures on the ASVAB are not necessarily drawn exactly to scale.
These two angles are complementary. Together they measure 90°.
Supplementary Angles Two adjacent angles whose measures total 180° are called supplementary angles. Together they make a straight line. KHG and GHJ are supplementary because together they add to 180° or a straight line.
The two angles below are supplementary. Together they measure 180° and form a straight line.
Vertical Angles Two angles formed by intersecting lines are called vertical angles if they are not adjacent. In the figure below, AED and BEC are vertical angles. AEB and DEC are also vertical angles. Vertical angles are often said to be "opposite" to each other, as shown in the figure.
Vertical angles are congruent. That is, their measures are the same. AED = BEC and AEB = DEC.
Identifying Congruent (Equal) Angles In the figure below, lines AC and DF are parallel. They are intersected by a third line GH. This third line is called a transversal.
This intersection creates eight angles. There are four pairs of vertical congruent angles:
ABH = EBC
ABE = HBC
DEB = GEF
DEG = BEF
Alternate Interior Angles In addition, four of these angles make two pairs of alternate interior angles. These are angles that are on opposite sides of the transversal, are between the two parallel lines, and are not adjacent. When parallel lines are intersected by a transversal, alternate interior angles are congruent. The two pairs are:
ABE = BEF
DEB = EBC
Alternate Exterior Angles Four of the angles also make two pairs of alternate exterior angles. These are angles that are on opposite sides of the transversal, are outside the two parallel lines, and are not adjacent. When parallel lines are intersected by a transversal, alternate exterior angles are congruent. The two pairs are:
ABH = GEF
HBC = DEG
Corresponding Angles Eight of the angles also make four pairs of corresponding angles. These are angles that are in corresponding positions. When parallel lines are intersected by a transversal, corresponding angles are congruent. The four pairs are:
ABH = DEB
HBC = BEF
ABE = DEG
EBC = GEF
HINT Angles count! Pay attention to these angle relationships! They are almost certain to appear in some form on the ASVAB.
Solving Angle Problems On the ASVAB, you will most likely be asked to use what you know about angles and angle relationships to solve problems. You may be asked to tell which angles in a figure are congruent. Or you may be given the measure of one angle and asked for the measure of an adjacent angle or some related angle in a figure.
In the following diagram, parallel lines MO and RT are intersected by transversal WV.
Which angle is congruent to MNW?
Of the choices, the only one that is congruent to MNW is VST because they are alternate exterior angles.
Which angle is congruent to MNS?
Of the choices, the only one that is congruent to MNS is RSV because they are corresponding angles.
If RSN measures 50°, what is the measure of RSV?
RSN and RSV are supplementary angles. That is, together they form a straight line and their measures add up to 180°. So if RSN measures 50°, then RSV measures 180 – 50 = 130°.
A polygon is a closed figure that can be drawn without lifting the pencil. It is made up of line segments (sides) that do not cross. A triangle is a polygon with three sides. Every triangle has three angles that total 180°.
Identifying the Longest Side of a Triangle The longest side of a triangle is always opposite the largest angle. So, if a triangle has angles of 45°, 55°, and 80°, the side opposite the 80° angle would be the longest.
Types of Triangles There are four main types of triangles. They are equilateral, isosceles, scalene, and right. Each has special characteristics that you should know.
Equilateral Triangle This kind of triangle has three congruent (equal) sides and three congruent (equal) angles. In an equilateral triangle, each angle measures 60°.
Isosceles Triangle This type of triangle has at least two congruent sides, and the angles opposite the congruent sides are also congruent. In the isosceles triangle shown below, sides AB and BC are congruent. BAC and BCA are also congruent. In an isosceles triangle, if you know the measure of any one angle, you can calculate the measures of the other two.
In this isosceles triangle, if 1 measures 30°, what is the measure of 3?
Since 1 and 2 are congruent, 2 must also measure 30°. Together, 1 and 2 add up to 60°. Since the sum of all three angles in any triangle is 180°, 3 must be 180 – 60 = 120°.
If 3 measures 100°, what are the measures of 1 and 2?
Since the sum of all three angles in any triangle is 180°, the sum of the measures of 1 and 2 must be 180 – 100 = 80°. Since angles 1 and 2 are congruent, each one must measure 80° ÷ 2 = 40°.
Scalene Triangle This kind of triangle has no equal sides or angles.
Right Triangle This kind of triangle has one angle that measures 90°. This angle is the right angle. It is identified in the figure by the little "box." Since the sum of all three angles in any triangle is 180°, the sum of the two remaining angles in a right triangle is 180 – 90 = 90°.
In a right triangle, there is a special relationship among the lengths of the three sides. This relationship is described by the Pythagorean Theorem.
In the right triangle below, C is the right angle. The side opposite the right angle is called the hypotenuse (c). It is always the longest side. The other two sides (a and b) are called legs.
According to the Pythagorean Theorem, in any right triangle, the sum of the squares of the legs equals the square of the hypotenuse. In symbols:
a2 + b2 = c2
So if you know the lengths of any two sides of a right triangle, you can calculate the length of the third side.
Base and Height of a Triangle Any side of a triangle can be called the base. The height is the length of a line segment that connects a base to the vertex opposite that base and is perpendicular to it.
Look at the triangle below. Dashed line CD is the height. Line CD is perpendicular to the base AB.
Where line CD meets base AB, it creates two right angles, CDA and CDB.
Median of a Triangle A median of a triangle is a line drawn from any vertex to the middle of the opposite side. This line splits the opposite side into two equal lengths.
Dashed line AD is a median of triangle ABC. It splits side BC into two equal lengths, .
A quadrilateral is a polygon with four sides and four angles. The sum of the four angles is always 360°.
Types of Quadrilaterals There are several different kinds of quadrilaterals. Each type is classified according to the relationships among its sides and angles. The square, rectangle, parallelogram, and rhombus are all types of quadrilaterals.
Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel and congruent. The opposite angles are also congruent. Around the edge of the parallelogram, each pair of consecutive angles is supplementary; that is, their sum is 180°. Diagonal lines drawn from opposite vertices bisect each other (divide each other exactly in half), but the diagonals themselves are not equal in length.
Like triangles, quadrilaterals have bases and height. Any side of this parallelogram can be a base. Dashed line is a height of this parallelogram. The height is a line originating at a vertex and drawn perpendicular to the opposite base. The height forms two right angles where it meets the base.
Rhombus A rhombus is a parallelogram with four congruent sides. Opposite angles are also congruent.
Rectangle A rectangle is a parallelogram with four right angles. Diagonal lines drawn from opposite vertices of a rectangle bisect each other (divide each other exactly in half) and are equal in length.
Square A square is a rectangle with four congruent sides. Diagonal lines drawn from opposite vertices of a rectangle bisect each other (divide each other exactly in half) and are equal in length.
Trapezoid A trapezoid is a quadrilateral with only one pair of parallel sides. Like other quadrilaterals, it has bases and height. In the example below, dashed line CE is the height. Sides AB and CD are parallel, but sides AC and BD are not parallel.
A circle is a closed figure with all points the same distance from a center. A circle with its center at point A is called circle A.
Parts of a Circle A chord is a line segment that has endpoints on a circle. A diameter is a chord that passes through the center of a circle. A radius is a line segment that connects the center of a circle and a point on the circle. Its length equals half the length of the diameter. In the figure below, A is the center of the circle. EF is a chord. BC is a diameter of the circle. AC and AB are each a radius of the circle.
An arc is two points on a circle and the part of the circle between the two points. In the figure below, CD is an arc of the circle. A central angle is an angle whose vertex is the center of a circle. In the figure below, CAD is a central angle. Its measure is 50°. The sum of the measures of the central angles in a circle is 360°.
Perimeter and Area
The perimeter is the distance around a closed twodimensional figure. The area is the amount of surface a two-dimensional figure covers. Area is measured in square units such as square inches (in2) or square centimeters (cm2). A square inch is the area of a square with sides 1 inch long.
Finding the Perimeter of a Polygon To find the perimeter of a polygon, just add the length of each side to find the total.
ΔABC is an isosceles triangle. If side AB has a length of 25 cm and side BC has a length of 15 cm, what is the perimeter?
Since ΔABC is an isosceles triangle, side AB = side AC. So if side AB has a length of 25 cm, side AC also has a length of 25 cm. Thus the perimeter is 25 + 25 + 15 = 65 cm.
Finding the Area of a Polygon There are special formulas you can use to calculate the areas of various types of polygons. You will want to memorize these, as you will almost certainly be asked a question about area on the ASVAB.
Area of a Triangle The area (A) of a triangle is onehalf the base (b) multiplied by the height (h), or
If a triangle has a height that measures 30 cm and a base that measures 50 cm, what is its area?
b = 50 cm
h = 30 cm
Area of a Square or Rectangle The area (A) of a square or rectangle is its length (l) multiplied by its width (w), or A = lw
If this rectangle is 10 miles long and 6 miles wide, what is its area?
A = lw
A = (10)(6)
A = 60 square miles (60 mi2)
Area of a Parallelogram (Including a Rhombus) The area (A) of a parallelogram is its base (b) multiplied by its height (h), or A = bh.
If a parallelogram has a base of 6 meters and a height of 4 meters, what is the area?
A = bh
A = (6)(4)
A = 24 square meters (24 m2)
Area of a Trapezoid The area (A) of a trapezoid is one-half the sum of the two bases (b1 and b2) multiplied by the height (h), or
If a trapezoid has one base of 30 meters and another base of 60 meters, and its height is 20 meters, what is the area?
Finding the Circumference and Area of a Circle The circumference of a circle is the distance around the circle. The circumference (C) divided by the diameter (d) always equals the number π (pi). Pi is an infinite decimal, meaning that its decimal digits go on forever. When you use it to solve problems, you can approximate as π 3.14 or 22/7.
To find the circumference of a circle, use the formula C = πd.
If a circle has a radius of 3 inches, what is the circumference?
Since the diameter is twice the radius (2r), the diameter is 6 inches.
C = πd
C = 3.14(d)
C = 3.14(6)
C = 18.84 in.
To find the area of a circle, multiply π times the square of the radius: A = πr2
If a circle has a radius of 4 centimeters, what is its area?
A = πr2
A = 3.14(4)2
A = 3.14(16)
A = 50.24 cm2
Three-Dimensional (Solid) Figures
A figure is two-dimensional if all the points on the figure are in the same plane. A square and a triangle are two-dimensional figures. A figure is three-dimensional (solid) if some points of the figure are in a different plane from other points in the figure.
On solid figures, the flat surfaces are called faces. Edges are line segments where two faces meet. A point where three or more edges intersect is called a vertex.
Types of Solid Figures On the ASVAB, you may see problems related to these solid figures: rectangular solid (prism), cube, cylinder, and sphere.
Rectangular Solid (Prism) On a rectangular solid (also called a prism), all of the faces are rectangular. The top and bottom faces are called bases. All opposite faces on a rectangular solid are parallel and congruent.
Cube A cube is a rectangular solid on which every face is a square.
Cylinder A cylinder is a solid figure with two parallel congruent circular bases and a curved surface connecting the boundaries of the two faces.
Sphere A sphere is a solid figure that is the set of all points that are the same distance from a given point, called the center. The distance from the center is the radius (r) of the sphere.
Finding the Volume of Solid Figures Volume is the amount of space within a three-dimensional figure. Volume is measured in cubic units, such as cubic inches (in3) or cubic centimeters (cm3). A cubic inch is the volume of a cube with edges 1 inch long.
Volume of a Rectangular Solid To find the volume (V) of a rectangular solid, multiply the length (l ) times the width (w) times the height (h). The formula is V = lwh.
If a rectangular solid has a length of 3 yards, a height of 1.5 yards, and a width of 1.5 yards, what is its volume?
V = lwh
V = (3)(1.5)(1.5)
V = 6.75 cubic yards (6.75 yd3)
Volume of a Cube On a cube, the length, width, and height are all the same: Each one equals 1 side (s). To find the volume (V) of a cube, multiply the length × width × height. This is the same as multiplying side × side × side. The formula is V = s × s × s = s3.
If each side of a cube measures 9 feet, what is its volume?
V = s3
V = (9)3
V = 729 cubic feet (729 ft3)
Volume of a Cylinder To find the volume (V) of a cylinder, first find the area of the circular base by using the formula A = πr2. Then multiply the result times the height (h) of the cylinder. The formula is V = (πr2)h).
If a cylinder has a height of 7 meters and a radius of 2 meters, what is its volume?
V = (πr2)h
V = 3.14(2)2(7)
V = 3.14(4)(7)
V = 87.92 cubic meters (87.92 m3)
Volume of a Sphere To find the volume of a sphere, multiply 4/3 times π times the radius cubed. The formula is
If the radius of a sphere measures 12 inches, what is the volume?
Geometry Word Problems
Word problems on the ASVAB may deal with geometry concepts such as perimeter, area, and volume. To solve these problems, you may also need to use information about different units of measure. To review units of measure, see Chapter 8.
Just as with other kinds of word problems, you can solve geometry problems by following a specific procedure. In the examples that follow, pay special attention to the procedure outlined in each solution. Follow this same procedure whenever you need to solve this kind of word problem.
Sally is buying wood to make a rectangular picture frame measuring 11 in. × 14 in. The wood costs 25 cents per inch. How much will Sally have to pay for the wood?
What must you find? Cost of the wood for the frame
What are the units? Dollars and cents
What do you know? Cost of the wood per inch; shape of the frame, measure of the frame
Create an equation and solve.
Length of wood needed for rectangular frame = 2l + 2w
Substitute values and solve:
Length of wood = 2(14) + 2(11)
Length of wood = 28 + 22 = 50 in.
If each inch costs 0.25, then
0.25 × 50 = $12.50
Sergei is planting rosebushes in a rectangular garden measuring 12 ft × 20 ft. Each rosebush needs 8 ft2 of space. How many rosebushes can Sergei plant in the garden?
What must you find? Number of rosebushes that can be planted in the garden
What are the units? Numbers
What do you know? Shape of the garden, garden length and width, amount of area needed for each rosebush
Create an equation and solve.
A = lw
Substitute values and solve.
A = 12 × 20 = 240 ft2
Each rosebush needs 8 ft2
240 ÷ 8 = 30
Sergei can plant 30 rosebushes in the garden.
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