# Lesson Explainer: Congruent Polygons Mathematics • 8th Grade

In this explainer, we will learn how to identify congruent polygons and use their properties to find a missing side length or angle.

Recall that polygons are two-dimensional shapes with straight sides.

Each point where two sides of a polygon meet is called a vertex (the plural is “vertices”).

Recalling also that congruent angles are angles that have the same measure and congruent sides are sides that have the same length, we can give a definition of congruent polygons as follows.

### Definition: Congruent Polygons

Two polygons are congruent if there is a correspondence between their vertices such that all corresponding interior angles and all corresponding sides are congruent.

Conversely, if two polygons are congruent, then there is a correspondence between their vertices such that all corresponding interior angles and all corresponding sides are congruent.

In other words, congruent polygons are polygons whose vertices and sides coincide exactly. Another way to think of this concept is that congruent polygons are polygons with the same shape and size, although they can be rotations, translations, or mirror images of each other.

In order to prove that two polygons are congruent, we need to show that

1. all corresponding sides are congruent, which means they have the same length;
2. all corresponding interior angles are congruent, which means they have the same measure.

On the other hand, if we are told that two polygons are congruent, this immediately implies that conditions (i) and (ii) must hold.

Note that the polygons with the smallest number of sides (three) are triangles. There are special rules for proving the congruency of triangles, and these are covered in another lesson. Here, we start with an example about a type of four-sided polygon: the square.

### Example 1: Congruence of Squares

Are two squares congruent if the side length of one square is equal to the side length of the other?

All squares have four vertices, so we can always form a correspondence between the vertices of one and the vertices of another. To prove that two squares with the same side length are congruent, we need to show that all corresponding interior angles are congruent and all corresponding sides are congruent.

We know that all squares have four equal interior angles of (i.e., four right angles).

This means that for every interior angle in one square, any corresponding interior angle in the other square must measure the same. Therefore, all corresponding interior angles are congruent.

Now, checking the sides, we are told that the two squares have the same side length, which we can label as . Since both squares have four sides of length , then for every side in one square, any corresponding side in the other square must have the same length.

Therefore, all corresponding sides are congruent.

Since we have shown that all corresponding interior angles and all corresponding sides are congruent, this implies that the two squares themselves are congruent.

We conclude that the answer to the question is yes, two squares are congruent if the side length of one is equal to the side length of the other.

Before moving on to more complicated problems, we introduce a helpful piece of mathematical notation.

For objects and , we write to mean that and are congruent.

For polygons and , the notation implies that the interior angle at vertex is congruent to the one at vertex , the interior angle at vertex is congruent to the one at vertex , and the interior angle at vertex is congruent to the one at vertex . Furthermore, side is congruent to side , side is congruent to side , and side is congruent to side .

The same labeling convention applies to all congruent polygons, irrespective of their number of sides. For example, for two four-sided polygons, we would use the notation .

By using this notation, we can express detailed information about the properties of congruent polygons in a very concise way. In particular, the order of the vertex letters tells us which interior angle is congruent to which and also which side is congruent to which. Our next example shows how to apply this knowledge.

### Example 2: Understanding the Notation for Congruence

The symbol means that the two objects are congruent. Which statement is true?

The diagram shows a four-sided polygon (or quadrilateral) split into two triangles that share the side . From the wording of the question, we know that one of the four answer options is correct, so we may assume the two triangles are congruent.

Recall that if two polygons are congruent, then there is a correspondence between their vertices such that all the corresponding interior angles and sides are congruent. By comparing the two triangles, we need to work out which interior angle is congruent to which. This will then enable us to use mathematical notation to describe the congruence relationship between the triangles, so that we can pick the correct answer option.

The three different side lengths in each triangle are marked with either a single dash, a double dash, or with no dashes (the shared side). In , the interior angle at vertex (written ) is between the shared side and the side with a single dash. Similarly, is between the sides with single and double dashes and is between the side with a double dash and the shared side.

In , tracking the interior angles in the same order by sides, we see that is between the shared side and the side with a single dash, is between the sides with single and double dashes, and is between the side with a double dash and the shared side.

These correspondences tell us that as shown in the diagram below.

We can express this congruence relationship by the notation , but this is not one of the four available answer options.

Consequently, it is important to remember that there is more than one way to describe the same congruence relationship, depending on the vertex we start at and the direction of travel around the polygon. For instance, instead of starting at vertex in , we could have started at or , so the following three statements are equivalent:

Additionally, if we travel around the polygon in the opposite direction, we get three more equivalent statements:

We have now listed all possible congruence relationships between the two triangles. The only one from this list that appears as an answer option is , so statement B is correct.

In the above example, we knew that the given triangles had a congruence relationship, but in many questions, we will be asked to check whether or not two polygons are congruent.

### Example 3: Identifying Congruent Polygons

Are the polygons shown congruent?

Recall that two polygons are congruent if there is a correspondence between their vertices such that all the corresponding interior angles and sides are congruent. Therefore, if we can show that these conditions are satisfied, then the polygons must be congruent.

From the diagram, the polygons and are both parallelograms, so in theory, we can form a correspondence between their vertices. Starting with vertex of parallelogram , it has an interior angle of . Comparing with parallelogram , we see that the only possible corresponding vertices are or , so

Repeating this step for vertices , , and of , we deduce that

Next, we compare side lengths. Starting with side of parallelogram , we see that its length is marked with a single dash. Comparing with parallelogram , we see that the only possible corresponding sides are or , so

Repeating this process for the other sides of , we get

This means we have a choice of correspondences, but to prove that the two parallelograms are congruent, it is sufficient to find one set of correspondences that works. Choosing implies that , , and . Therefore, our answer is yes, the two polygons are congruent, with . We can see this more clearly if we rotate the polygon , as shown in the diagram below.

Note that if we had chosen instead, it would follow that , , and . Again, the two polygons would be congruent, but this time with . This can be seen by rotating the polygon as below.

Once we have identified two polygons as being congruent, we can sometimes use their properties to find a missing side length or angle in geometric problems. Let’s look at an example of this type.

### Example 4: Finding the Measure of an Angle Bounded between Two Congruent Quadrilaterals

Given that , find the measure of .

Recall that congruent polygons are the same shape and size, but they can be rotations, translations, or mirror images of each other. We are told that , and from the diagram, we see that polygon is actually a reflection of polygon in the straight line that passes through , perpendicular to the line segment .

As with , we deduce that . Therefore, we know two of the three angles at above the line segment . The missing angle is , which we have been asked to work out. Recalling the fact that angles on a straight line sum to , we have the equation

Subtracting and from both sides gives and substituting the values and , we get

Thus, we have calculated that .

In our final example, we apply the properties of congruent polygons in a geometric context.

### Example 5: Using the Properties of Congruence to Solve a Geometry Problem

The perimeter of the polygon is 176 cm and . Given that and , find the perimeter of the figure .

Recall that congruent polygons are the same shape and size, but they can be rotations, translations, or mirror images of each other. The question states that , and from the diagram, we see that polygon is a reflection of polygon in the straight line containing the line segment .

We have been asked to find the perimeter of the figure , which is the new shape formed from the two original polygons by excluding the shared side , as shown below.

In the question, the notation tells us that is a line segment, so the horizontal sides of polygon and of polygon join to make the single horizontal side of figure .

To calculate the perimeter of , we need to add together the perimeters of and , but in both cases, we must exclude the length of the side (we write this length as just ). Then, writing for the perimeter of and so on, we have

From the question, the perimeter of the polygon is 176 cm, with . This implies that the perimeter of the polygon is also 176 cm, so substituting these values into the above equation, we get

Finally, we know that , so substituting this value gives

All the lengths were given in centimeters, so the perimeter of the figure is 256 cm.

Let us finish by recapping some key concepts from this explainer.

### Key Points

• Two polygons are congruent if there is a correspondence between their vertices such that all corresponding interior angles and all corresponding sides are congruent. Conversely, if two polygons are congruent, then there is a correspondence between their vertices such that all corresponding interior angles and all corresponding sides are congruent.
• Congruent polygons are polygons with the same shape and size, but they can be rotations, translations, or mirror images of each other.
• For objects and , we write to mean that and are congruent. For polygons and , the notation implies that the interior angle at vertex is congruent to the one at vertex , the interior angle at vertex is congruent to the one at vertex , and the interior angle at vertex is congruent to the one at vertex. Furthermore, side is congruent to side , side is congruent to side , and side is congruent to side .
• The same labeling convention applies to all congruent polygons, irrespective of their number of sides. For example, for two four-sided polygons, we would use the notation .
• We can use the properties of congruent polygons to work out a missing side length or angle in geometric problems.