Geometry Concepts: GED Test Prep (page 3)
This article reviews the geometry concepts you will need to know for the GED Mathematics Exam. You should become familiar with the properties of angles, lines, polygons, triangles, and circles, as well as the formulas for area, volume, and perimeter. A grasp of coordinate geometry will also be important when you take the GED.
Geometry is the study of shapes and the relationships among them. The geometry you are required to know for the GED Mathematics Exam is fundamental and practical. Basic concepts in geometry will be detailed and applied in this section. The study of geometry always begins with a look at basic vocabulary and concepts. Therefore, here is a list of definitions of important terms.
- area—the space inside a two-dimensional figure
- bisect—cut in two equal parts
- circumference—the distance around a circle
- diameter—a line segment that goes directly through the center of a circle (the longest line you can draw in a circle)
- equidistant—exactly in the middle of
- hypotenuse—the longest leg of a right triangle, always opposite the right angle
- line—an infinite collection of points in a straight path
- point—a location in space
- parallel—lines in the same plane that will never intersect
- perimeter—the distance around a figure
- perpendicular—two lines that intersect to form 90-degree angles
- quadrilateral—any four-sided closed figure
- radius—a line from the center of a circle to a point on the circle (half of the diameter)
- volume—the space inside a three-dimensional figure
An angle is formed by an endpoint, or vertex, and two rays.
There are three ways to name an angle.
- An angle can be named by the vertex when no other angles share the same vertex: A.
- An angle can be represented by a number written across from the vertex: 1.
- 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 as DAB; 2 can be written as DAC or as CAD.
Angles can be classified into the following categories: acute, right, obtuse, and straight.
- An acute angle is an angle that measures less than 90 degrees.
- A right angle is an angle that measures exactly 90 degrees. A right angle is represented by a square at the vertex.
- An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees.
- A straight angle is an angle that measures 180 degrees. Thus, both of its sides form a line.
Two angles are complementary if the sum of their measures is equal to 90 degrees.
Two angles are supplementary if the sum of their measures is equal to 180 degrees.
Adjacent angles have the same vertex, share a side, and do not overlap.
The sum of all of the measures of adjacent angles around the same vertex is equal to 360 degrees.
Angles of Intersecting Lines
When two lines intersect, two sets of nonadjacent angles called vertical angles are formed. Vertical angles have equal measures and are supplementary to adjacent angles.
- m1 = m3 and m2 = m4
- m1 = m4 and m3 = m2
- m1 + m2 = 180 and m2 + m3 = 180
- m3 + m4 = 180 and m1 + m4 = 180
Bisecting Angles and Line Segments
Both angles and lines are said to be bisected when divided into two parts with equal measures.
Line segment AB is bisected at point C.
According to the figure, A is bisected by ray AC.
Angles Formed by Parallel Lines
- When two parallel lines are intersected by a third line, vertical angles are formed.
- Of these vertical angles, four will be equal and acute, and four will be equal and obtuse.
- Any combination of an acute and an obtuse angle will be supplementary.
In the previous figure:
- b, c, f, and g are all acute and equal.
- a, d, e, and h are all obtuse and equal.
- Also, any acute angle added to any obtuse angle will be supplementary.
mb + md = 180°
mc + me = 180°
mf + mh = 180°
mg + ma = 180°
In the following figure, if m||n and a||b, what is the value of x?
Because both sets of lines are parallel, you know that x° can be added to x + 10 to equal 180. The equation is thus x + x + 10 = 180.
Solve for x:
Therefore, mx = 85 and the obtuse angle is equal to 180 – 85 = 95.
Angles of a Triangle
The measures of the three angles in a triangle always equal 180 degrees.
An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some rules for working with exterior angles:
- An exterior angle and interior angle that share the same vertex are supplementary.
- An exterior angle is equal to the sum of the nonadjacent interior angles.
m1 = m3 + m5
m4 = m2 + m5
m6 = m3 + m2
The sum of the exterior angles of a triangle equal 360 degrees.
It is possible to classify triangles into three categories based on the number of equal sides:
It is also possible to classify triangles into three categories based on the measure of the greatest angle:
Knowing the angle-side relationships in isosceles, equilateral, and right triangles will be useful in taking the GED Exam.
- In isosceles triangles, equal angles are opposite equal sides.
- In equilateral triangles, all sides are equal and all angles are equal.
- In a right triangle, the side opposite the right angle is called the hypotenuse. This will be the largest side of the right triangle.
The Pythagorean theorem is an important tool for working with right triangles. It states a2+ b2= c2, where a and b represent the legs and c represents the hypotenuse.
This theorem allows you to find the length of any side as long as you know the measure of the other two.
45–45–90 Right Triangles
A right triangle with two angles each measuring 45 degrees is called an isosceles right triangle. In an isosceles right triangle:
- The length of the hypotenuse is multiplied by the length of one of the legs of the triangle.
- The length of each leg is multiplied by the length of the hypotenuse.
In a right triangle with the other angles measuring 30 and 60 degrees:
- The leg opposite the 30-degree angle is half of the length of the hypotenuse. (And, therefore, the hypotenuse is two times the length of the leg opposite the 30-degree angle.)
- The leg opposite the 60-degree angle is times the length of the other leg.
Triangles are said to be congruent (indicated by the symbol ≅) when they have exactly the same size and shape. Two triangles are congruent if their corresponding parts (their angles and sides) are congruent. Sometimes, it is easy to tell if two triangles are congruent by looking. However, in geometry, you must be able to prove that the triangles are congruent.
If two triangles are congruent, one of the following three criteria must be satisfied.
- Side-Side-Side (SSS)—The side measures for both triangles are the same.
- Side-Angle-Side (SAS)—The sides and the angle between them are the same.
- Angle-Side-Angle (ASA)—Two angles and the side between them are the same
Example: Are ΔABC and ΔBCD congruent?
Given: ABD is congruent to CBD and ADB is congruent to CDB. Both triangles share side BD.
Step 1: Mark the given congruencies on the drawing.
Step 2: Determine whether this is enough information to prove the triangles are congruent.
Yes, two angles and the side between them are equal. Using the ASA rule, you can determine that triangle ABD is congruent to triangle CBD.
Polygons and Parallelograms
A polygon is a closed figure with three or more sides.
Terms Related to Polygons
Vertices are corner points, also called endpoints, of a polygon. The vertices in this polygon are A, B, C, D, E, and F.
A diagonal of a polygon is a line segment between two nonadjacent vertices. The two diagonals in this polygon are line segments BF and AE.
- A regular polygon has sides and angles that are all equal.
- An equiangular polygon has angles that are all equal.
Angles of a Quadrilateral
A quadrilateral is a four-sided polygon. Because a quadrilateral can be divided by a diagonal into two triangles, the sum of its interior angles will equal 180 + 180 = 360 degrees.
To find the sum of the interior angles of any polygon, use this formula:
S = 180(x – 2)°,with x being the number of polygon sides
Find the sum of the angles in the following polygon:
S = (5 – 2) × 180°
S = 3 × 180°
S = 540°
Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees.
If two polygons are similar, their corresponding angles are equal and the ratios of the corresponding sides are in proportion.
These two polygons are similar because their angles are equal and the ratios of the corresponding sides are in proportion.
A parallelogram is a quadrilateral with two pairs of parallel sides.
In this figure, line AB || CD and BC || AD.
A parallelogram has:
- opposite sides that are equal (AB = CD and BC = AD)
- opposite angles that are equal (ma = mc and mb = md)
- and consecutive angles that are supplementary (ma + mb = 180°, mb + mc = 180°, mc + md = 180°,md + ma = 180°)
Special Types of Parallelograms
- A rectangle is a parallelogram that has four right angles.
- A rhombus is a parallelogram that has four equal sides.
- A square is a parallelogram in which all angles are equal to 90 degrees and all sides are equal to each other.
In all parallelograms, diagonals cut each other into two equal halves.
- In a rectangle, diagonals are the same length.
- In a rhombus, diagonals intersect to form 90-degree angles.
- In a square, diagonals have both the same length and intersect at 90-degree angles.
Solid Figures, Perimeter, and Area
The GED provides you with several geometrical formulas. These formulas will be listed and explained in this section. It is important that you be able to recognize the figures by their names and to understand when to use which formulas. Don't worry. You do not have to memorize these formulas. They will be provided for you on the exam.
To begin, it is necessary to explain five kinds of measurement:
- Surface Area
The perimeter of an object is simply the sum of all of its sides.
Area is the space inside of the lines defining the shape.
Volume is a measurement of a three-dimensional object such as a cube or a rectangular solid. An easy way to envision volume is to think about filling an object with water. The volume measures how much water can fit inside.
The surface area of an object measures the area of each of its faces. The total surface area of a rectangular solid is double the sum of the areas of the three faces. For a cube, simply multiply the surface area of one of its sides by six.
Circumference is the measure of the distance around the outside of a circle.
Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A coordinate plane is a grid of square boxes divided into four quadrants by both a horizontal (x) axis and a vertical (y) axis. These two axes intersect at one coordinate point, (0,0), the origin. A coordinate point, also called an ordered pair, is a specific point on the coordinate plane with the first number, or coordinate, representing the horizontal placement and the second number, or coordinate, representing the vertical. Coordinate points are given in the form of (x, y).
Graphing Ordered Pairs
- The x-coordinate is listed first in the ordered pair and tells you how many units to move to either the left or to the right. If the x-coordinate is positive, move to the right. If the x-coordinate is negative, move to the left.
- The y-coordinate is listed second and tells you how many units to move up or down. If the y-coordinate is positive, move up. If the y-coordinate is negative, move down.
Graph the following points: (2,3), (3,–2), (–2,3), and (–3,–2).
Notice that the graph is broken up into four quadrants with one point plotted in each one.
Here is a chart to indicate which quadrants contain which ordered pairs based on their signs:
Lengths of Horizontal and Vertical Segments
Two points with the same y-coordinate lie on the same horizontal line and two points with the same x-coordinate lie on the same vertical line. The length of a horizontal or vertical segment can be found by taking the absolute value of the difference of the two points, or by counting the spaces on the graph between them.
Find the length of line AB and line BC.
To find the midpoint of a segment, use the following formula:
Midpoint x =
Midpoint y =
Find the midpoint of line segment AB.