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Properties of Congruence

Identify corresponding angles and sides

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Recognizing and Understanding Congruent Polygons
License: CC BY-NC 3.0

A geodesic dome is a structure that looks like a half sphere and is made up of triangle supports. Geodesic domes are efficient and inexpensive to construct and therefore are used in home and commercial construction. Are the triangles in a geodesic dome congruent?

In this concept, you will learn all about the importance of congruence and how to determine congruence.

Congruent Polygons

What does it mean when you say that two figures are congruent? The word congruent means exactly the same. When you have two figures of any kind that have the same size, shape and measure, you can say that these two figures are congruent.

Let’s look at an example.

Are these two angles congruent?

License: CC BY-NC 3.0

If you look at these two angles, you will see that they are both 55°. The angles are labeled that way, so you can see that they are equal. This means that they are congruent. You can say that angle \begin{align*}A\end{align*} is congruent to angle \begin{align*}B\end{align*}.

The answer is \begin{align*}\angle A \cong \angle B\end{align*}.

Just as you said that any two figures can be congruent, you can use this when you look at different types of polygons too. Two polygons can be considered congruent or not congruent.

Let’s look at an example involving congruence.

Are these two octagons congruent?

License: CC BY-NC 3.0

Look at these two octagons. They are exactly the same in every way. You can see that if you put one octagon on top of the other octagon that they would match up perfectly. The side lengths are also congruent and the angle measures are congruent. If two polygons are congruent, then it is a given that the side lengths and the angle measures are also congruent.

Now that you know how to identify whether or not two figures are congruent, you can look at figuring out congruent parts and angles.

First, let’s think again about the main characteristics of congruent polygons:

  1. Same number of sides
  2. Corresponding sides are the same length
  3. Corresponding interior angles are the same measure

The last two characteristics can be a bit tricky. Sometimes, you will have two congruent figures, but all of the measures of the corresponding angles won’t be the same. For example, if you had two irregular congruent hexagons, this means that there are different angle measures in the two hexagons; however, they are congruent so there are “matching” angles between the two figures.

Take a look at these figures.

License: CC BY-NC 3.0

Here you have two hexagons. They are irregular - which means that all of the side lengths and angles are not the same. However, they are congruent. You can see that one matches the other. Because of this, you have corresponding angles that are the same from the first hexagon to the second hexagon.

You can identify corresponding parts of congruent figures. Corresponding parts can include side lengths and angle measures.

Let’s look at an example.

Name each pair of corresponding side lengths for these congruent figures.

License: CC BY-NC 3.0

First, look at these two congruent pentagons. To name the corresponding sides, you name the sides that match from one pentagon to another pentagon.

Next, write the pairs of congruent sides. Here are the corresponding sides and how you can write them using mathematical notation.

\begin{align*}\begin{array}{rcl} \overline{BA} & \cong & \overline {GF}\\ \overline{BC} & \cong & \overline {GH}\\ \overline{CD} & \cong & \overline {HK}\\ \overline{AE} & \cong & \overline {FL}\\ \overline{ED} & \cong & \overline {LK} \end{array}\end{align*}

The answer is \begin{align*}\overline{BA} \cong \overline {GF}, \overline{BC} \cong \overline {GH}, \overline{CD} \cong \overline {HK}, \overline{AE} \cong \overline {FL}, \overline{ED} \cong \overline {LK}\end{align*}.

Then, look at the angles. You can also look at the corresponding angles for two congruent figures. When two figures are congruent, then the matching angles will also be congruent.

Examples

Example 1

Earlier, you were given a problem about a geodesic dome.

First, let’s think about what the word congruent means. Congruent means exactly the same. For an object to be congruent, the side lengths have to be the same. The triangles in the geodesic dome have to be congruent for it to stand up because the triangle is a structure that is well balanced to help with structure and security. Triangles are used in all kinds of construction like roofs and bridges.

The test for congruence of the triangles would be to see that the side lengths are the same.

Example 2

Are these two hexagons congruent?

License: CC BY-NC 3.0

First, look at these two shapes and compare them.

One shape is a regular hexagon and the other is an irregular hexagon.

Next, use the congruent polygon properties to see if the two hexagons are congruent.

In order to be congruent, corresponding sides must be the same length. The regular hexagon has all side lengths with the same measure. The irregular hexagon does not have sides that are the same length.

The answer is no, the two hexagons are not congruent as their corresponding sides are not all the same measure.

Use the following figures to answer each question.

License: CC BY-NC 3.0

Example 3

What is the angle measure of angle \begin{align*}F\end{align*}?

First, find the corresponding angle in quadrilateral \begin{align*}ABCD\end{align*}.

\begin{align*}\angle F \cong \angle B\end{align*}

Next, find the measure of the corresponding angle. This will be the measure of \begin{align*}\angle F\end{align*}.

The answer is 100°.

Example 4

Angle \begin{align*}D\end{align*} is congruent to which other two angles?

First, find the corresponding angle in quadrilateral \begin{align*}EFGH\end{align*}.

\begin{align*}\angle D \cong \angle H\end{align*}

Next, look at quadrilateral \begin{align*}ABCD\end{align*}. What are the measures of all of angles?

\begin{align*}\begin{array}{rcl} \angle A &=& 80^\circ\\ \angle B &=& 100^\circ\\ \angle C &=& 80^\circ\\ \angle D &=& 100^\circ \end{array}\end{align*}

Then, if \begin{align*}\angle B = \angle D \text{ and } \angle B \cong \angle F, \text{ then } \angle D \cong \angle H \cong \angle F\end{align*}.

The answer is \begin{align*}\angle D \cong \angle H \cong \angle F\end{align*}.

Example 5

What is the measure of angle \begin{align*}G\end{align*}?

First, find the corresponding angle in quadrilateral \begin{align*}ABCD\end{align*}.

\begin{align*}\angle G \cong \angle C\end{align*}

Next, find the measure of the corresponding angle. This will be the measure of \begin{align*}\angle G\end{align*}.

The answer is 80°.

Review

Answer each question true or false.

1. Congruent means that a figure has the same side lengths but not the same angle measures.

2. Congruent means exactly the same in every measure.

3. Similar means having the same shape, but not the same size.

4. Two congruent figures would have the same size and shape.

5. Corresponding parts are parts that are in the same figure.

6. You need to understand corresponding parts before you can determine if two figures are congruent.

7. You can determine if two figures are congruent without knowing any of their measurements.

8. Similar figures are also congruent.

9. If two triangles are equilateral triangles, then they are automatically congruent.

10. If two quadrilaterals have measures of 360°, then they are congruent.

The two figures shown are congruent. Use the illustration to answer each question.

License: CC BY-NC 3.0

11. If angle \begin{align*}B\end{align*} has a measure of 75°, which other angle has the same measure?

12. If angle \begin{align*}F\end{align*} is 120°, which other angle has the same measure?

13. True or false. Angle \begin{align*}E\end{align*} and angle \begin{align*}K\end{align*} have the same measure.

14. True or false. Angle \begin{align*}C\end{align*} and angle \begin{align*}H\end{align*} have the same measure.

15. Name this figure.

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.10. 

Resources

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Vocabulary

Congruent

Congruent figures are identical in size, shape and measure.

Corresponding Angles

Corresponding angles are two angles that are in the same position with respect to the transversal, but on different lines.

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