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8.12: Congruent Figures

Difficulty Level: At Grade Created by: CK-12
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Mrs. Gilman brought a small group of students over to look at this tile floor in the hallway of the art museum.

“You see, there is even math in the floor,” she said smiling. Mrs. Gilman is one of those teachers who loves to point out every place where math can be found.

“Okay, I get it,” Jesse started. “I see the squares.”

“There is a lot more math than just squares,” Mrs. Gilman said walking away with a huge smile on her face.

“She frustrates me sometimes,” Kara whispered staring at the floor. “Where is the math besides the squares?”

“I think she is talking about the size of the squares,” Hannah chimed in. “See there are two different sizes.”

“Actually there are three different sizes and there could be more that I haven’t found yet,” Jesse said.

“Remember when we learned about comparing shapes that are alike and aren’t alike, it has to do with proportions or something like that,” Hannah chimed in again.

All three students stopped talking and began looking at the floor again.

“Oh yeah, congruent and similar figures, but which are which?”Kara asked.

"Let's start with the congruent ones," Hannah said.

What are congruent figures? This Concept will teach you all about congruent figures. When you are all finished with this Concept, you will have a chance to study the floor again and see if you can find the congruent figures.

Guidance

The word congruent means exactly the same. Sometimes, you will see this symbol \begin{align*}\cong\end{align*}.

In this Concept, we are going to use the word congruent to compare figures.

Congruent figures have exactly the same size and shape. They have congruent sides and congruent angles. Here are some pairs of congruent figures.

Compare the figures in each pair. They are exactly the same! If you’re not sure, imagine that you could cut out one figure and place it on top of the other. If they match exactly, they are congruent.

How can we recognize congruence?

We test for congruency by comparing each side and angle of two figures to see if all aspects of both are the same. If the sides are the same length and the angles are equal, the figures are congruent. Each side and angle of one figure corresponds to a side or angle in the other. We call these corresponding parts. For instance, the top point of one triangle corresponds to the top point of the other triangle in a congruent pair.

It is not always easy to see the corresponding parts of two figures. One figure may be rotated differently so that the corresponding parts appear to be in different places. If you’re not sure, trace one figure and place it on top of the other to see if you can make them match. Let’s see if we can recognize some congruent figures.

Which pair of figures below is congruent?

Let’s analyze one pair at a time to see if we can find any corresponding angles and sides that are congruent.

The figures in the first pair appear to be the same shape, if we rotate one \begin{align*}180^\circ\end{align*}180 so they both point up. Now we can see all of the corresponding parts, such as the angle at the top and the two long sides on the left and right. This is not enough to go on, however. We need to compare the measures of the angles and the lengths of the sides. If any one set of corresponding parts doesn’t match, the figures cannot be congruent.

We only know the measure of one angle in the first two figures. We can compare these angles if they are corresponding parts. They are, because if we rotate one figure these angles are in the same place at the top of each figure. Now compare their measures. The angle in the first figure is \begin{align*}45^\circ\end{align*}45. The corresponding angle in the second figure is \begin{align*}55^\circ\end{align*}55. Because the angles are different, these two figures are not congruent. Let’s look at the next pair.

The two triangles in the second pair seem to have corresponding parts: a long base and a wide angle at the top. We need to know whether any of these corresponding parts are congruent, however. We know the measure of the top angle in each figure: it is \begin{align*}110^\circ\end{align*}110 in both. These figures might be congruent, but we need to see if the sides are congruent to be sure (as we said, similar figures also have congruent angles, but their sides are different lengths). We know the measure of each triangle’s base: one is 2 inches and the other is 4 inches. These sides are not congruent, so the triangles are not congruent. Remember, every side and every angle must be the same in order for figures to be congruent.

That leaves the last pair. Can you find the corresponding parts? If we rotate the second figure \begin{align*}180^\circ\end{align*}180, we have two shapes that look like the letter \begin{align*}L\end{align*}. Now compare the corresponding sides. The bottom side of each is 8 cm, the long left side of each is 8 cm, two sides are 6 cm, and two sides are 2 cm. All of the angles in both figures are \begin{align*}90^\circ\end{align*}. Because every side and angle in one figure corresponds to a congruent side and angle in the second, these two figures are congruent.

What about angle measures of congruent figures?

We know that congruent figures have exactly the same angles and sides. That means we can use the information we have about one figure in a pair of congruent figures to find the measure of a corresponding angle or side in the other figure. Let’s see how this works. Take a look at the congruent figures below.

We have been told these two parallelograms are congruent.

Can you find the corresponding parts?

If not, trace one parallelogram and place it on top of the other. Rotate it until the parts correspond.

Which sides and angles correspond?

We can see that side \begin{align*}AB\end{align*} corresponds to side \begin{align*}PQ\end{align*}. Because they are congruent, we write.

\begin{align*}AB \cong PQ.\end{align*}

What other sides are congruent? Let’s write them out.

\begin{align*}AB &\cong PQ\\ BC &\cong QR\\ AD &\cong PS\\ DC &\cong SR\end{align*}

We can also write down the corresponding angles, which we know must be congruent because the figures are congruent.

\begin{align*}\angle A &\cong \angle P && \angle D \cong \angle S\\ \angle B &\cong \angle Q && \angle C \cong \angle R\end{align*}

Now that we understand all of the corresponding relationships in the two figures, we can use what we know about one figure to find the measure of a side or angle in the second figure.

Can we find the length of side \begin{align*}AB\end{align*}?

We do not know the length of \begin{align*}AB\end{align*}. However, we do know that it is congruent to \begin{align*}PQ\end{align*}, so if we can find the length of \begin{align*}PQ\end{align*} then it will be the same for \begin{align*}AB. PQ\end{align*} is 7 centimeters. Therefore \begin{align*}AB\end{align*} must also be 7 centimeters long.

Now let’s look at the angles. Can we find the measure of \begin{align*}\angle C\end{align*}?

It corresponds to \begin{align*}\angle R\end{align*}, but we do not know the measure of \begin{align*}\angle R\end{align*} either. Well, we do know the measures of two of the angles in the first parallelogram: \begin{align*}70^\circ\end{align*} and \begin{align*}110^\circ\end{align*}. If we had three, we could subtract from \begin{align*}360^\circ\end{align*} to find the fourth, because all quadrilaterals have angles that add up to \begin{align*}360^\circ\end{align*}. We do not know the measure of \begin{align*}\angle B\end{align*}, but this time we do know the measure of its corresponding angle, \begin{align*} \angle Q\end{align*}. These two angles are congruent, so we know that \begin{align*}\angle B\end{align*} must measure \begin{align*}70^\circ\end{align*}. Now we know three of the angles in the first figure, so we can subtract to find the measure of \begin{align*}\angle C\end{align*}.

\begin{align*}360 - (70 + 110 + 70) &= \angle C\\ 360 - 250 &= \angle C\\ 110^\circ &= \angle C\end{align*}

We were able to combine the given information from both figures because we knew that they were congruent.

Yes and the more you work on puzzles like this one the easier they will become.

Now it's time for you to try a few on your own.

Example A

True or false. Congruent figures have the same number of sides and angles.

Solution: True

Example B

True or false. Congruent figures can have one pair of angles with the same measure, but not all angles have the same measure.

Solution: False

Example C

True or false. Congruent figures can be different sizes as long as the angle measures are the same.

Solution: False

Now back to the congruent figures at the art museum.

Mrs. Gilman brought a small group of students over to look at this tile floor in the hallway of the art museum.

“You see, there is even math in the floor,” she said smiling. Mrs. Gilman is one of those teachers who loves to point out every place where math can be found.

“Okay, I get it,” Jesse started. “I see the squares.”

“There is a lot more math than just squares,” Mrs. Gilman said walking away with a huge smile on her face.

“She frustrates me sometimes,” Kara whispered staring at the floor. “Where is the math besides the squares?”

“I think she is talking about the size of the squares,” Hannah chimed in. “See there are two different sizes.”

“Actually there are three different sizes and there could be more that I haven’t found yet,” Jesse said.

“Remember when we learned about comparing shapes that are alike and aren’t alike, it has to do with proportions or something like that,” Hannah chimed in again.

All three students stopped talking and began looking at the floor again.

“Oh yeah, congruent and similar figures, but which are which?”Kara asked.

"Let's start with the congruent ones," Hannah said.

The congruent figures are exactly the same. We can say that the small dark brown squares are congruent because they are just like each other. They have the same side lengths. What is one other pair of congruent squares?

Make a note of the congruent figures that you can find and then share your findings with a friend. Compare answers and continue with the Concept.

Vocabulary

Here are the vocabulary words in this Concept.

Congruent
having exactly the same shape and size. All side lengths and angle measures are the same.

Guided Practice

Here is one for you to try on your own.

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

Answer

We can use reasoning to figure out this problem. First, we know that the two triangles are congruent. We also know two of the three angle measures of the triangles.

Let's write an equation.

\begin{align*}95 + 35 + x = 180\end{align*}

Now we can solve for the missing angle measure.

\begin{align*}130 + x = 180\end{align*}

\begin{align*}x = 50\end{align*}

The measure of the missing angle is \begin{align*}50^\circ\end{align*}.

Video Review

Here is a video for review.

- This is a James Sousa video on similar and congruent triangles.

Practice

Directions: Name the corresponding parts to those given below.

1. \begin{align*}\angle R\end{align*}

2. \begin{align*}MN\end{align*}

3. \begin{align*}\angle O\end{align*}

Directions: Use the relationships between congruent figures to find the measure of \begin{align*}g\end{align*}. Show your work.

4.

Directions: Use the relationships between congruent figures to find the measure of \begin{align*}\angle T\end{align*}. Show your work.

5.

Directions: Answer each of the following questions.

6. Triangles \begin{align*}ABC\end{align*} and \begin{align*}DEF\end{align*} are congruent. If the measure of angle \begin{align*}A\end{align*} is \begin{align*}58^{\circ}\end{align*}, what is the measure of angle \begin{align*}D\end{align*} if it corresponds to angle \begin{align*}A\end{align*}?

7. True or false. Congruent figures are exactly the same in every way.

Directions: Identify the given triangles as visually congruent or not.

8.

9.

10.

11.

12.

Directions: Answer each of the following questions.

13. Triangles \begin{align*}ABC\end{align*} and \begin{align*}DEF\end{align*} are congruent. Does this mean that their angle measures are the same? Why?

14.Define Congruent.

15. True or false. If two figures are congruent, then they have the same length sides but not the same angle measures.

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Difficulty Level:
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Date Created:
Nov 30, 2012
Last Modified:
Feb 26, 2016
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