Mathematics Study Guide for the TABE Test

Geometry: Part 2

We have been very specific about lines and angles in preparation for this geometry section. Lines and angles are the building blocks of most shapes.

Shapes

Much of geometry is focused on measuring and manipulating shapes. You will need to know the names of common shapes, particularly flat shapes, and their properties.

Two-Dimensional

Two-dimensional (\(2\)-D) shapes are flat shapes, known in geometry as plane figures. All this means is that every one of these shapes exists entirely on a flat surface (a plane). They only have two dimensions, length and width, with no thickness rising out of the flat surface.

Common 2-D Shapes

There are many types of two-dimensional shapes, some that you will see often in math classes and real life, others that are more unusual. All of them are defined by specific properties that we will discuss in the next sections. These are examples of some of the most common types of shapes:

All of the shapes with straight edges are polygons. The circles are a different category, and we’ll discuss them in greater depth later.

Properties of Quadrilaterals

A quadrilateral is any polygon with four sides. Quadrilaterals are especially important in geometry because they appear frequently in both math problems and in the real world. Rooms, windows, screens, and plots of land all tend to be quadrilaterals. Because of this, many TABE problems involving area, perimeter, and angles are based on quadrilaterals. Understanding their properties will make it much easier to solve these types of questions.

All quadrilaterals share a few key properties:

• They have four sides and four vertices (or corners).
• The sum of their interior angles is always \(360^\circ\).
• They can be divided into two triangles (which explains why the angle sum is \(360^\circ\)).

While all quadrilaterals have these basic features in common, different types of quadrilaterals have additional properties that make them unique:

• A rectangle has four right angles and opposite sides are equal and parallel.
• A square is a special type of rectangle in which all four sides are equal.
• A rhombus also has four equal sides, but its angles are not necessarily right angles.

Note: It’s possible for a shape to fall under more than one category. For instance, all squares are also rectangles, although not all rectangles are squares.

Some quadrilaterals are a bit less familiar but still very important. In a parallelogram, both pairs of opposite sides are parallel. In addition, opposite sides are equal in length and opposite angles are equal. A trapezoid has at least one pair of parallel sides, but unlike parallelograms, the other pair of sides may not be parallel, which gives trapezoids a different shape and set of properties.

Are there quadrilaterals that don’t fit into any of the discussed categories? Yes, you could draw any number of them. Here’s an example:

There are countless irregular shapes in the world, some with many sides. No matter how odd they may look, though, they still have to follow the basic rules of geometry.

Properties of Other Shapes

There are a number of other common shapes that you will see that are not quadrilaterals. This is a quick summary of some of the most important ones:

• A triangle has three sides and three interior angles that add up to \(180^\circ\).
• A pentagon has five sides and five interior angles that add up to \(540^\circ\).
• A hexagon has six sides and six interior angles that add up to \(720^\circ\).

A list of shapes could go on and on, but these are the basics.

If you’re noticing a pattern, that’s because there is one. The number of sides determines the number of interior angles. Also, with each additional interior angle, the total angle measurement increases by \(180^\circ\).

If you know how many sides or interior angles there are in a shape, you can determine the total angle measurement with this formula:

\[180(n-2)\]

where \(n\) is the number of interior angles. For instance, if a shape has \(10\) interior angles, the measure of all its interior angles is:

\[180(10-2) = 180(8) = 1\text{,}440\]

Comparing 2-D Shapes

There are two useful methods for comparing shapes. One way is called similarity. Two figures are similar if they have the same shape, but not necessarily the same size. For example, all squares are similar, though some are big and some are small. The two pentagons below are similar. All the angles are the same, but the sides of one aren’t the same lengths as the sides of the other:

The other common way to compare shapes is congruency. Two shapes are congruent if all the angles are the same in both shapes and all the matching sides have the same length. A good way to think about it is to notice that one figure will fit exactly on top of the other. These two right triangles are congruent:

Shapes can be similar or congruent even if they don’t have the same orientation. For instance, if the second triangle above was flipped onto its shorter side, it would still be congruent with the first triangle.

Angle-Angle Similarity

When working with triangles, there is a very useful shortcut for determining whether two triangles are similar. This is known as angle-angle similarity, often shortened to AA similarity.

The rule is simple:

If two angles in one triangle are equal to two angles in another triangle, then the triangles are similar.

This works because the sum of the angles in any triangle is always \(180^\circ\). So, if two angles match, the third angle must also match automatically. Let’s do an example.

Both Triangle A and Triangle B have two angles measuring \(50^\circ\) and \(60^\circ\). Are they similar? If so, what’s the measure of the third angle?

Solution

Based on the AA similarity rule, since two angles of both the triangles are the same, Triangle A and Triangle B are similar triangles. Moreover, the two given angles add up to \(50+60 =110^{\circ}\). Thus, the third angle will measure:

\[180 - 110 = 70^{\circ}\]

Even if the triangles are different sizes, they will still have the same shape. This means:

• Their corresponding angles are equal.
• Their corresponding sides are proportional (they follow the same ratio).

Angle-angle similarity is especially helpful because you do not need to measure all three sides or angles to compare triangles. Just two matching angles are enough to confirm similarity. This makes it a quick and reliable method when solving geometry problems.

Three-Dimensional

As mentioned earlier, plane figures are two-dimensional shapes that have no thickness. On the other hand, there are three-dimensional (\(3\)-D) shapes, also known as solid shapes, that do have thickness. Put another way, these shapes have depth (or height) as well as length and width.

Common 3-D Shapes

There are several common three-dimensional shapes that you will encounter on the TABE. Each has its own structure and properties, but many of them are related in useful ways.

A prism is a solid shape that has:

• two parallel, congruent bases (top and bottom)
• rectangular faces connecting those bases

The shape of the base determines the type of prism. For example, a rectangular prism has rectangular bases and a triangular prism has triangular bases.

A right prism is a prism in which the side faces meet the bases at right angles (\(90^\circ\)). This makes the sides stand straight up, rather than leaning.

A rectangular prism (also called a box shape) has:

• rectangular faces
• opposite faces that are equal

A rectangular prism may or may not be a right prism.

A cube is a special type of rectangular prism in which:

• all faces are squares
• all edges are equal in length

A cylinder has:

• two parallel circular bases
• one curved surface connecting them

A pyramid has:

• one base (which can be any polygon)
• triangular faces that meet at a single point called the apex

A cone has:

• one circular base
• one curved surface that comes to a point (apex)

A sphere is a perfectly round solid shape where:

• all points on the surface are the same distance from the center
• it has no flat faces, edges, or vertices

Attributes of 3-D Shapes

With two-dimensional shapes, we have sides made of line segments. With certain solids, such as cubes and pyramids, we refer to the flat surfaces as faces. The line segments where the faces meet are called edges. The point where two edges meet is called a vertex, similar to a vertex in a two-dimensional shape.

The cube above has six faces, twelve edges, and eight vertices. Not all solids will have all of these attributes. For instance, a sphere doesn’t have faces, edges, or vertices because there are no flat surfaces.

Working with Shapes

Manipulating shapes is the business of geometry. In this section we will take a look at dividing, combining, moving, and measuring them.

Partitioning

Shapes can be partitioned (or divided) into smaller shapes with equal areas that are fractions of the original shape. For example, a square can be divided into four parts with equal areas, as such:

Each part is one-fourth of the original square. The whole square is made of four fourths (\(\frac{4}{4}\)).

Composing

As opposed to dividing shapes, we can put shapes together (compose them) to form new shapes. Below, you can see that we started with a circle and a rectangle. Then the circle was split into two semicircles. The last step was to compose the three shapes into a new shape:

Decomposing

Decomposing is the process of breaking a complex shape into smaller and simpler shapes that are easier to work with. While partitioning usually divides shapes into equal parts, decomposing focuses on splitting a shape in a way that helps us measure or analyze it more easily.

For example, an irregular shape can often be broken down into rectangles, triangles, or circles. Once the shape is separated into these familiar parts, you can calculate measurements such as area or perimeter for each piece and then combine the results. Decomposing is especially useful when dealing with real-world figures, such as floor plans, plots of land, or composite objects, where shapes are not always neat or uniform.

Measuring Complex Shapes

The knowledge of composing and decomposing is especially helpful when finding the area of complex shapes. Instead of trying to apply a single formula, you can break the figure into smaller, manageable parts. Once that is done, you can find the area of each and then combine them. Let’s try an example.

Find the area of the figure shown below:

Solution

There are two ways to solve this problem. We can break apart the figure into two rectangles, find the area of each, and add them. Or we can recognize that the bottom-right part is a rectangular chunk taken out of the whole rectangle. We would then subtract that from the whole to get the result. We will take the first approach by breaking apart the figure like so:

Now, we find the area of rectangle \(1\). The length and width is \(6\), so the area is:

\[6 \times 6 = 36\]

Next, we find the area of rectangle \(2\). The width is \(3\) and the length is \(10 - 6 = 4\). So, the area is:

\[3 \times 4 = 12\]

Thus, the total area of the figure is \(36 + 12 = 48\) square units.

If you want to find the area in the second way, first we have to find the area of the whole rectangle, as if it didn’t have the bottom-right part taken out. So, based on the two full sides, the area is:

\[10 \times 6 = 60\]

The bottom-right rectangle has an area of \(3 \times 4 = 12\). Now, we subtract this from the whole to find the area of the figure:

\[60 - 12 = 48\]

As you can see, we got the same answer as we did above.

Manipulating Shapes

Shapes can be moved around in different ways called transformations, some of which are shown below. In each case the original shape is shown in solid lines, and the new transformed shape is drawn in dotted lines. These are the forms of transformation you should know:

• Translation is the sliding or moving of the original shape without spinning it:

• Rotation is spinning a shape around a point. In the example below, the flag is spinning around point \(O\). The point can be on the shape or, as in this case, outside the shape:

• Reflection means flipping the shape over a line. See in the below image how the top triangle appears to be pointing down, but the bottom one is pointing up:

• Dilation is expanding or shrinking an image while keeping the shape the same:

You can do more than one transformation on a given shape, such as rotating, then translating, and then dilating. Even when all of that is done, though, the shape will still be fundamentally the same shape.