GED Mathematical Reasoning: Four Sided Plane Figures

Rectangle  

A rectangle is a four-sided figure containing four right angles, each measuring 90 degrees. Opposite sides are the same lengths and are parallel. Notice the markings I’ve placed on the figure. A small square in each corner is used to denote the angle measure of 90 degrees. And hash marks are used to denote side length where sides with the same markings are the same length.

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Square

Just like in a rectangle, a square contains four right angles and opposite sides are parallel. The square has the added property that all four sides are the same length.

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Parallelogram

In a parallelogram, opposite sides are parallel and of equal length. In addition, opposite angles – those diagonally across from one another – have the same measure. Notice that each of the four corners is labeled with a letter. Using those letters, we can refer to this figure as parallelogram ABCD.

In addition, we can use the letters when referencing a specific side. For instance, we might say that side AB and side DC are parallel.

Furthermore, we can use the letters as names for the four angles. For example angles A and C have the same measure. And angles B and D have the same measure.

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Trapezoid

A trapezoid is four-sided figure with exactly one pair of parallel sides. There are no additional parameters given for a trapezoid with respect to the length of the sides or measure of the angles. In the example showing on the screen, Sides AB and DC are parallel.

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Although these four figures look different and have different characteristics, there is one unifying property that is true of all four. The sum of all the angles in any four-sided plane figure is 360 degrees.

This unifying property, along with the individual characteristics of each figure, can help us determine missing angle measures.

 

Example 1

In figure EFGH, opposite sides are parallel. If \angle H measures 50^{\circ}, find the measure of the other angles.

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When solving problems involving plane figures, a good first step is to identify the figure and review its properties.

We are told in the directions that opposite sides of the figure shown are parallel. Furthermore, the hash marks on the figure denote that opposite sides are the same lengths. Therefore, the figure showing in this example is a parallelogram.

Recall that in a parallelogram, opposite angles – those diagonally across from one another – have the same measure. In terms of our example, this means that angles H and F have the same measure and angles E and G have the same measure.

We are given that angle H measures 50 degrees, which means that angle F also measures 50 degrees.

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From here, we can set up and solve an equation to find the measure of the remaining angles, angles E and G.

So let’s let x equal the measure of angle E. Since the measure of angle G is the same as the measure of angle E, then x represents the measure of angle G, as well.

Let x = \angle E = \angle G

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Since the sum of all four angles must be 360 degrees, the equation that represents this situation is:

\angle E + \angle F + \angle G + \angle H = 360

x + 50 + x + 50 = 360

Now let’s solve!

2x + 100 = 360

2x = 260

x = 130

The solution is: x = 130

Since x represents the measure of both angles E and G, the measure of each of these angles is 130 degrees.

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We can do a quick check of our work by adding up all the angle measures to ensure the sum equals 360 degrees. Does 50 plus 50 plus 130 plus 130 equal 360? Yes! So we can be confident in our result.

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