The Fundamentals of Geometry Study Guide (page 3)
Introduction to The Fundamentals of Geometry
The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.
—ARISTOTLE, Greek philosopher (384 b.c.e.–322 b.c.e.)
Geometry typically represents only a small portion of most standardized math tests. The geometry questions that are included tend to cover the basics: points, lines, planes, angles, triangles, rectangles, squares, and circles. You may be asked to determine the area or perimeter of a particular shape, the size of an angle, the length of a line, and so forth. Some word problems may also involve geometry. And, as the word problems will show, geometry problems come up in real life as well.
Points, Lines, and Planes
What Is a Point?
A point has position but no size or dimension. It is usually represented by a dot named with an uppercase letter:
What Is a Line?
A line consists of an infinite number of points that extend endlessly in both directions.
A line can be named in two ways:
- By a letter at one end (typically in lowercase): l
- By two points on the line:
The following terminology is frequently used on math tests:
||Points are collinear if they lie on the same line. Points J,U,D, and I are collinear.|
||A line segment is a section of a line with two endpoints. The line segment at right is indicated as .|
||The midpoint is a point on a line segment that divides it into two line segments of equal length. M is the midpoint of line segment .|
||Two line segments of the same length are said to be congruent. Congruent line segments are indicated by the same mark on each line segment. are congruent. are congruent. Because each pair of congruent line segments is marked differently, the four segments are NOT congruent to each other.|
||A line segment (or line) that divides another line segment into two congruent line segments is said to bisect it. bisects .|
What Is a Plane?
A plane is like a flat surface with no thickness. Although a plane extends endlessly in all directions, it is usually represented by a four-sided figure and named by an uppercase letter in a corner of the plane: K.
Points are coplanar if they lie on the same plane.
Points A and B are coplanar.
What Is an Angle?
An angle is formed when two lines meet at a point: The lines are called the sides of the angle, and the point is called the vertex of the angle.
The symbol used to indicate an angle is
There are three ways to name an angle:
By the letter that labels the vertex:
- By the three letters that label the angle: , with the vertex letter in the middle
- By the number inside the vertex:
An angle's size is based on the opening between its sides. Size is measured in degrees (°). The smaller the angle, the fewer degrees it has. Angles are classified by size. Notice how the are shows which of two angles is indicated:
Acute angle: less than 90°
Right angle: exactly 90°
The little box indicates a right angle. A right angle is formed by two perpendicular lines. (Perpendicular lines are discussed at the end of the lesson.)
Straight angle: exactly 180°
Obtuse angle: more than 90°and less than 180°
Since bisect means to cut into two congruent pieces, an angle bisector cuts an angle into two equal angles.
is angle bisector to
When two angles have the same degree measure, they are said to be congruent.
Congruent angles are marked the same way.
The symbol is used to indicate that two angles are congruent:.
Complementary, Supplementary, and Vertical Angles
Names are given to three special angle pairs, based on their relationship to each other:
- Complementary angles: Two angles whose sum is 90°.
are complementary angles.
is the complement of , and vice versa.
- Supplementary angles: Two angles whose sum is 180°.
are supplementary angles.
is the supplement of , and vice versa.
Hook: To prevent confusing complementary and supplementary:
C comes before S in the alphabet, and 90 comes before 180.Complementary = 90° Supplementary = 180°
Sometimes more than two angles are combined to form a straight angle. When this occurs, the sum of all the angles is 180°.
- Vertical angles: Two angles that are opposite each other when two lines cross. Two sets of vertical angles are formed:
Vertical angles are congruent.
When two lines cross, the adjacent angles are supplementary and the sum of all four angles is 360°.
Angle-pair problems tend to ask for an angle's complement or supplement.
|Example:||If = 35°, what is the size of its complement?|
|To find an angle's complement, subtract it from 90°:||90°– 35°= 55°|
|Check: Add the angles to be sure their sum is 90°.|
|Example:||If = 35°, what is the size of its supplement?|
|To find an angle's supplement, subtract it from 180°:||180°– 35°= 145°|
|Check: Add the angles to be sure their sum is 180°.|
|Example:||If = 70°, what are the sizes of the other three angles?|
|1.||because they're vertical angles.|
|Therefore, = 70°.|
|2.||and are adjacent angles and therefore supplementary.|
|Thus, = 110°(180°– 70°= 110°).|
|3.||because they're also vertical angles.|
|Therefore, = 110°.|
|Check: Add the angles to be sure their sum is 360°.|
To solve geometry problems more easily, draw a picture if one is not provided. Try to draw the picture to scale. If the problem presents information about the size of an angle or line segment, label the corresponding part of your picture to reflect the given information. As you begin to find the missing information, label your picture accordingly.
Special Line Pairs
Parallel lines lie in the same plane and don't cross at any point:
The arrowheads on the lines indicate that they are parallel. The symbol is used to indicate that two lines are parallel: l m.
A transversal is a line that crosses two parallel lines. Line t is a transversal.
When two parallel lines are crossed by a transversal, two groups of four angles each are formed. One group consists of ,, , and ; the other group contains , , , and .
The angles formed by the transversal crossing the parallel lines have special relationships:
- The four obtuse angles are congruent:
- The four acute angles are congruent:
- The sum of any one acute angle and any one obtuse angle is 180°because the acute angles lie on the same line as the obtuse angles.
|Hook:||As a memory trick, draw two parallel lines and cross them with a transversal at a very slant angle. All you have to remember is that there will be exactly two sizes of angles. Since half the angles will be very small and the other half will be very large, it should be clear which ones are congruent. The placement of the congruent angles will be the same on every pair of parallel lines crossed by a transversal.|
Perpendicular lines lie in the same plane and cross to form four right angles.
|The little box where the lines cross indicates a right angle. Because vertical angles are equal, and the sum of all four angles is 360°, each of the four angles is a right angle. However, only one little box is needed to indicate this.|
|The symbol is used to indicate that two lines are perpendicular:|
|Don't be fooled into thinking two lines are perpendicular just because they look perpendicular. The problem must indicate the presence of a right angle (by stating that an angle measures 90°or by the little right angle box in a corresponding diagram), or you must be able to prove the presence of a 90°angle.|
Look for acute, right, obtuse, and straight angles throughout the day. For instance, check out a bookcase. What is the size of the angle formed by the shelf and the side of the bookcase? Is it an acute, obtuse, or right angle? Imagine that you could bend the side of the bookcase with your bare hands. How many degrees would you have to bend it to create an acute angle? How about a straight angle? Take out a book and open it. Form the covers into an acute angle, a right angle, or a straight angle.
Find practice problems and solutions for these concepts at The Fundamentals of Geometry Practice Questions.