Mathematics Study Guide for the TABE Test

Geometry: Part 1

Geometric concepts are often woven into questions on fractions, percentage, percent change, and ratios, among other math and number concepts. TABE word problems will often involve planes and solid geometric shapes and require you to be familiar with these concepts:

• points in a coordinate plane
• lines and slopes
• angles
• polygons
• area and perimeter of polygons
• the Pythagorean theorem
• circles
• area and circumference of circles
• volume of solids

In the following sections, we will cover these important geometry concepts.

Points and Lines

When you think of a point, you probably picture some kind of little spot or dot, and that idea works pretty well in geometry too. Technically, a point is an infinitely small location, but we can’t actually put that on paper, so we just put a small dot down and call it a point. For the sake of differentiation, points are named with a capital letter.

A line in geometry is something that is straight, infinitely thin, and infinitely long. Being infinitely thin, we can’t see a true line, so we just draw something straight and designate it as a line. A line has no endpoints; instead, we add arrowheads at either end of the line to show that it continues forever in both directions. Lines are usually named using two points that are on the line.

The line below would be named line \(AB\) and the symbol for it is \(\overleftrightarrow{AB}\):

A line segment is a finite piece of a line that is capped on both ends by two endpoints. The piece of the line above between points \(A\) and \(B\) would be named segment \(AB\), and the symbol for it is \(\overline{AB}\).

A ray is kind of like half a line. It has one endpoint and continues infinitely in one direction, like a laser. A ray is named using two points, always starting with the endpoint. The ray below would be named ray \(AB\), and its symbol is \(\overrightarrow{AB}\).

Perpendicular and Parallel Lines

Perpendicular lines (or segments) meet to form four right angles, which are angles that are \(90\) degrees (\(90^\circ\)). You’ll learn more about angles and degrees shortly, but just know that each of these angles is shaped like a capital L.

Below, line \(AB\) is perpendicular to line \(CD\):

Using symbols, this would be written \(\overleftrightarrow{AB} \perp \overleftrightarrow{CD}\). The little square where the lines meet is the symbol for a right angle. Although not specifically marked, the other three angles there are right angles too. (You’ll see why when you learn about supplementary angles shortly.)

Parallel lines are lines in the same plane that never meet, no matter how far they are extended. They are drawn like this:

We could say that line \(AB\) is parallel to line \(CD\), or, in symbols, we could write \(\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}\).

Angles

An angle is formed when two rays or line segments share a common endpoint, known as the vertex (vertices is the plural). Angles are named using the angle symbol (\(\angle\)) and three letters, one from each side and the vertex point. The vertex point is always in the middle, but the order of the other two points doesn’t matter. The right angle in the next section could be named \(\angle CAR\) or \(\angle RAC\).

Types of Angles

There are three main types of angles, which are named depending on their degrees, a unit of measurement used to describe the width of an angle:

• Right angles have measures of exactly \(90^\circ\).
• Acute angles are less than \(90^\circ\).
• Obtuse angles are greater than \(90^\circ\).

Adjacent Angles

In the two figures below, you will see examples of adjacent angles. They share a vertex and a common side between them. Two adjacent angles that add up to \(90^\circ\) are said to be complementary, so \(\angle ABD\) and \(\angle DBC\) in the figure below are a pair of complementary angles:

Two adjacent angles that add up to \(180^\circ\) are said to be supplementary angles, so in the figure below, \(\angle CAT\) and \(\angle TAR\) are a pair of supplementary angles:

An angle can also be named by putting a number between the two sides near the vertex, as below. Keep in mind that these numbers do not represent degrees. This image shows two pairs of supplementary angles:

In the above figure you can see that when two lines intersect they form four angles. The pairs of angles that are opposite from each other are called vertical angles (in this case, vertical doesn’t mean straight up). So, \(\angle 1\) and \(\angle 3\) are a pair of vertical angles, and so are \(\angle 2\) and \(\angle 4\). Vertical angles are always congruent to each other, meaning they have the same measure in degrees. We could write \(\angle 1 \cong \angle 3\) and \(\angle 2 \cong \angle 4\).

Also, any straight line that is split by a ray forms a pair of supplementary angles. For example, \(\angle 1\) and \(\angle 2\) are supplementary angles.

Based on these properties, you will be required to solve geometric problems. Let’s look at an example problem using the last figure.

If \(\angle 1 = 60^\circ\), what is the sum of \(\angle 2\) and \(\angle 4\)?

Solution

You know that \(\angle 1\) and \(\angle 2\) are supplementary, so \(\angle 1 + \angle 2 = 180^\circ\). Since \(\angle 1\) is \(60^\circ\), \(\angle 2\) must be \(120^\circ\) to make a sum of \(180^\circ\).

Furthermore, \(\angle 2\) and \(\angle 4\) are vertical angles, so they are congruent and \(\angle 2 = \angle 4\). That means \(\angle 4=120^\circ\).

Therefore, we add them together to get \(120+120=240^\circ\).

Transversal Cuts

When a line crosses two other lines, it is called a transversal. Transversals are especially important when working with parallel lines, because they create several pairs of angles with special relationships. The following angle relationships are formed:

• corresponding angles—These angles are in the same relative position and are equal.

• alternate interior angles—These angles are on opposite sides of the transversal and inside the parallel lines; they are equal.

• alternate exterior angles—These are on opposite sides of the transversal and outside the parallel lines; they are also equal.

• same-side interior angles—These are inside the parallel lines on the same side of the transversal and are supplementary (they add up to \(180^\circ\)).

• same-side exterior angles—These are outside the parallel lines on the same side of the transversal and are also supplementary.

In the above figure, there are four pairs of corresponding angles:

• \(\angle a\) and \(\angle b\)
• \(\angle c\) and \(\angle d\)
• \(\angle e\) and \(\angle f\)
• \(\angle g\) and \(\angle h\)

There are two pairs of alternate interior angles:

• \(\angle e\) and \(\angle d\)
• \(\angle g\) and \(\angle b\)

There are two pairs of alternate exterior angles:

• \(\angle a\) and \(\angle h\)
• \(\angle c\) and \(\angle f\)

There are two pairs of same-side interior angles:

• \(\angle e\) and \(\angle b\)
• \(\angle g\) and \(\angle d\)

There are two pairs of same-side exterior angles:

• \(\angle a\) and \(\angle f\)
• \(\angle c\) and \(\angle h\)