8.5: Similar and Congruent Figures
Introduction
The Mathematical Floor
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.
What is the difference between congruent and similar figures? This lesson will teach you all about congruent and similar figures. When you are all finished with this lesson, you will have a chance to study the floor again and see if you can find the congruent and the similar figures.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills:
- Recognize congruence.
- Find unknown measures of congruent figures.
- Recognize similarity.
- Check for similarity between given figures
Teaching Time
I. Recognize Congruence
In the last lesson we began using the word “congruent.” We talked about congruent lines and congruent angles. The word congruent means exactly the same. Sometimes, you will see this symbol \begin{align*}\cong\end{align*}.
In this lesson, 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.
Example
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*} 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*}. The corresponding angle in the second figure is \begin{align*}55^\circ\end{align*}. 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*} 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*}, 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.
8J. Lesson Exercises
Answer true or false for each question
- Congruent figures have the same number of sides and angles.
- Congruent figures can have one pair of angles with the same measure, but not all angles have the same measure.
- Congruent figures can be different sizes as long as the angle measures are the same.
Discuss your answers with a friend. Be sure you understand why each answer is true or false.
II. Find Unknown 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.\end{align*} Since \begin{align*}PQ\end{align*} is 7 centimeters, \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.
8K. Lesson Exercises
Answer this question.
1. What is the measure of \begin{align*}\angle M\end{align*}? (The two triangles are congruent.)
Take a few minutes to check your answer with a friend. Correct any errors and then continue with the next section.
III. Recognize Similarity
Some figures look identical except they are different sizes. The angles even look the same. When we have figures that are proportional to each other, we call these figures similar figures. Similar figures have the same angle measures but different side lengths.
What is an example of similar figures?
Squares are similar shapes because they always have four \begin{align*}90^\circ\end{align*} angles and four equal sides, even if the lengths of their sides differ. Other shapes can be similar too, if their angles are equal.
Let’s look at some pairs of similar shapes.
Notice that in each pair the figures look the same, but one is smaller than the other. Since they are not the same size, they are not congruent. However, they have the same angles, so they are similar.
IV. Check for Similarity between Given Figures
Unlike congruent figures, similar figures are not exactly the same. They do have corresponding features, but only their corresponding angles are congruent; the corresponding sides are not. Thus when we are dealing with pairs of similar figures, we should look at the angles rather than the sides. In similar figures, the angles are congruent, even if the sides are not.
Notice that one angle in each pair of figures corresponds to an angle in the other figure. They have the same shape but not the same size. Therefore they are similar.
Let’s find the corresponding angles in similar figures.
Example
List the corresponding angles in the figures below.
Angles \begin{align*}G\end{align*} and \begin{align*}W\end{align*} are both right angles, so they correspond to each other. Imagine you can turn the figures to line up the right angles. You might even trace the small figure so that you can place it on top of the larger one.
How do the angles line up?
Angles \begin{align*}H\end{align*} and \begin{align*}X\end{align*} correspond to each other. So do angles \begin{align*}I\end{align*} and \begin{align*}Y\end{align*} and angles \begin{align*}J\end{align*} and \begin{align*}Z\end{align*}. Now we can name these two quadrilaterals: \begin{align*}GHIJ\end{align*} is similar to \begin{align*}WXYZ\end{align*}.
As we’ve said, the sides in similar figures are not congruent. They are proportional, however. Proportions have the same ratio. Look at \begin{align*}GHIJ\end{align*} and \begin{align*}WXYZ\end{align*} again. We can write each pair of sides as a proportion.
\begin{align*}\frac{GH}{WX},\frac{HI}{XY},\frac{IJ}{YZ},\frac{GJ}{WZ}\end{align*}
The sides from one figure are on the top, and the proportional sides of the other figure are on the bottom.
Example
List all of the pairs of corresponding sides in the figures below as proportions.
Try lining up the figures by their angles. It may help to trace one figure and rotate it until it matches the other.
Which sides are proportional?
\begin{align*}OP\end{align*} and \begin{align*}RS\end{align*} are the shortest sides in each figure. They are proportional, so we write
\begin{align*}\frac{OP}{RS}\end{align*}
Now that we’ve got one pair, let’s do the same for the rest.
\begin{align*}\frac{NO}{QR}, \frac{MP}{TS}, \frac{MN}{TQ}\end{align*}
Now let’s use what we have learned to check for similarity between figures.
Example
Which pair of figures below is similar?
For figures to be similar, we know that the angles must be congruent and the sides must exist in proportional relationships to each other. Let’s check each pair one at a time.
We only know some of the angles in each triangle in the first pair. They both have a \begin{align*}50^{\circ}\end{align*} angle, so that’s a good start. All three angles must be congruent, however, so let’s solve for the missing angle in each angle. Remember, the sum of the three angles is always \begin{align*}180^{\circ}\end{align*} for a triangle.
\begin{align*}&\text{Triangle 1} && \text{Triangle 2}\\ &50 + 60 + \text{angle} \ 3 = 180 && 50 + 80 + \text{angle} \ 3 = 180\\ &110 + \text{angle} \ 3 = 180 && 130 + \text{angle} \ 3 = 180\\ &\text{angle} \ 3 = 180 - 110 && \text{angle} \ 3 = 180 - 130 \\ &\text{angle} \ 3 = 70^{\circ} && \text{angle} \ 3 = 50^{\circ}\end{align*}
The angles in the first triangle are \begin{align*}50^{\circ}\end{align*}, \begin{align*}60^{\circ}\end{align*}, and \begin{align*}70^{\circ}\end{align*}. The angles in the second triangle are \begin{align*}50^{\circ}\end{align*}, \begin{align*}50^{\circ}\end{align*}, and \begin{align*}80^{\circ}\end{align*}. These triangles are not similar because their angle measures are different.
Let’s move on to the next pair.
This time we know side lengths, not angles. We need to check whether each set of corresponding sides is proportional. First, let’s write out the pairs of proportional corresponding sides
\begin{align*}\frac{6}{3}, \frac{6}{3}, \frac{4}{1}\end{align*}
The proportions show side lengths from the large triangle on the top and its corresponding side in the small triangle on the bottom. The pairs of sides must have the same proportion in order for the triangles to be similar. We can test whether the three proportions above are the same by dividing each. If the quotient is the same, the pairs of sides must exist in the same proportion to each other.
\begin{align*}\frac{6}{3} = 2\\ \frac{6}{3} = 2\\ \frac{4}{1} = 4\end{align*}
When we divide, only two pairs of sides have the same proportion (2). The third pair of sides does not exist in the same proportion as the other two, so these triangles cannot be similar.
That leaves the last pair. We have been given the measures of some of the angles. If all of the corresponding angles are congruent, then these two figures are similar. We know the measure of three angles in each figure. In fact, they are all corresponding angles. Therefore the one unknown angle in the first figure corresponds to the unknown angle in the second figure.
As we know, the four angles in a quadrilateral must have a sum of \begin{align*}360^{\circ}\end{align*}. Therefore the unknown angle in each figure must combine with the other three to have this sum. Because the three known angles are the same for both figures, we don’t even need to solve for the fourth to know that it will be the same in both figures. These two figures are similar because their angle measures are all congruent.
Now let’s use what we have learned to solve the problem in the introduction.
Real Life Example Completed
The Mathematical Floor
Here is the original problem once again. Reread it and then answer the questions at the end of this passage.
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.
The students are working on which figures in the floor pattern are congruent and which ones are similar.
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?
The similar figures compare squares of different sizes. You can see that the figures are squares, so they all have 90 degree angles. The side lengths are different, but because the angles are congruent, we can say that they have the same shape, but not the same size. This makes them similar figures.
The small dark brown square is similar to the large dark brown square. The small dark brown square is also similar to the square created by the ivory colored tile. There is a relationship between the different squares. Are there any more comparisons? Make a few notes in your notebook.
Vocabulary
- Congruent
- having exactly the same shape and size. All side lengths and angle measures are the same.
- Similar
- having the same shape but not the same size. All angle measures are the same, but side lengths are not.
Technology Integration
Khan Academy Congruent and Similar Triangles
James Sousa, Congruent and Similar Triangles
Time to Practice
Directions: Tell whether the pairs of figures below are congruent, similar, or neither.
Directions: Name the corresponding parts to those given below.
7. \begin{align*}\angle R\end{align*}
8. \begin{align*}MN\end{align*}
9. \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.
10.
Directions: Use the relationships between congruent figures to find the measure of \begin{align*}\angle T\end{align*}. Show your work.
11.
Directions: Answer each of the following questions.
12. 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*}?
13. True or false. If triangles \begin{align*}DEF\end{align*} and \begin{align*}GHI\end{align*} are similar, then the side lengths may be different but the angle measures are the same.
14. True or false. Similar figures have exactly the same size and shape.
15. True or false. Congruent figures are exactly the same in every way.
16. Triangles \begin{align*}LMN\end{align*} and \begin{align*}HIJ\end{align*} are similar. If this is true, then the side lengths are the same, true or false.
17. What is a proportion?
18. True or false. To figure out if two figures are similar, see if their side lengths form a proportion.
19. Define similar figures
20. Define congruent figures.