9.6: Congruent and Similar Figures
Introduction
The Pair of Skateboard Ramps
Marc, Isaac and Isabelle thought that designing a skateboard ramp would be easy. Because of this, they have decided to build two of them in their skatepark. Using the computer, they found the measurements for the first skateboard ramp design.
It has the form of a triangle and is in three dimensions, so it also has a width. Here are the dimensions for the first ramp.
28” long \begin{align*}\times\end{align*} 38.5” wide \begin{align*}\times\end{align*} 12” high
Isaac writes the following proportion on a piece of paper.
\begin{align*}\frac{28”}{14”} = \frac{38.5”}{\Box} = \frac{12”}{6”}\end{align*}
“The two ramps are going to be similar, but not congruent,” Isaac begins to explain.
At that moment, his mom begins calling him and he dashes out the door leaving Isabelle and Marc with his work and with the proportion.
“What is the difference between similar and congruent?” Isabelle asks.
“I am not sure,” Marc says. “But he didn’t finish the measurement either. What is the width of the second ramp?”
Isaac has left Isabelle and Marc with a problem. Can you help them figure this out? What is the difference between figures that are similar versus congruent? What is the missing dimension?
\begin{align*}\frac{28”}{14”} = \frac{38.5”}{} = \frac{12”}{6”}\end{align*}
What You Will Learn
By the end of this lesson, you will learn the following skills:
- Identify given triangles as similar, congruent or neither.
- Identify corresponding parts of congruent figures.
- Identify corresponding parts of similar figures.
- Find unknown measures of corresponding parts of similar figures.
- Use similar figures to measure indirectly.
Teaching Time
I. Identify Given Triangles as Similar, Congruent or Neither
You have heard the word congruent used regarding line segments being the same length. The word congruent can apply to other things in geometry besides lines and line segments. Congruent means being exactly the same. When two line segments have the same length, we can say that they are congruent. When two figures have the same shape and size, we can say that the two figures are congruent.
Example
These two triangles are congruent. They are exactly the same in every way. They are the same size and the same shape. We can also say that their side lengths are the same and that their angle measures are the same.
Sometimes, two figures will be similar. Similar means that the figures have the same shape, but not the same size. Similar figures are not congruent.
Example
These two triangles are similar. They are the same shape, but they are not the same size.
Identify the following triangles as congruent, similar or neither.
1.
2.
3.
Take a few minutes to check your work with a peer.
II. Identify Corresponding Parts of Congruent Figures
Now that you understand the difference between congruent figures and similar figures, we can look at the corresponding parts of congruent triangles. The word corresponding refers to parts that match between two congruent triangles. We can identify corresponding angles and corresponding sides.
Let’s look at an example.
Example
First, we can name the corresponding angles. Corresponding angles are matching angles between the two triangles. Corresponding angles will have the same measure in congruent triangles.
\begin{align*}\angle{A} \cong \angle{D} \\ \angle{B} \cong \angle{E} \\ \angle{C} \cong \angle{F}\end{align*}
Here the angles are connected with the symbol for congruent. When you see the equals sign with a squiggly line on top, you know that the items on each side of the equation are congruent.
Next, we can name the corresponding sides. Corresponding sides are matching sides between two triangles. They will have the same length in congruent triangles.
\begin{align*}\overline{AB} \cong \overline{DE} \\ \overline{AC} \cong \overline{DF} \\ \overline{BC} \cong \overline{EF}\end{align*}
Use the following diagram of two congruent triangles to answer each question.
- Angle \begin{align*}E\end{align*} is congruent to angle _____
- \begin{align*}\overline{FG} \cong \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
- Angle \begin{align*}J\end{align*} is congruent to angle _____
Take a few minutes to check your answers. Did you correctly match up the corresponding parts?
III. Identify Corresponding Parts of Similar Figures
We just finished identifying the corresponding parts of congruent figures, and we can also identify the corresponding parts of similar figures. We do this in the same way. Let’s look at an example.
Example
Triangle \begin{align*}ABC\end{align*} is similar to triangle \begin{align*}DEF\end{align*}. This means that while they are the same shape, they aren’t the same size. In fact, there is a relationship between the corresponding parts of the triangle.
The side lengths are corresponding even though they aren’t congruent.
\begin{align*}\overline{AB} \times \overline{DE} \\ \overline{BC} \times \overline{EF} \\ \overline{AC} \times \overline{DF}\end{align*}
We use the symbol for similar ("~") to show the relationship between the corresponding sides of the two triangles.
IV. Find Unknown Measures of Corresponding Parts of Similar Figures
Once you know how to locate the corresponding sides of similar triangles, we can write ratios to compare the lengths of sides. Let’s look at an example.
Example
First, identify the corresponding sides of these two similar triangles.
\begin{align*}\frac{LM}{OP} = \frac{LN}{OQ} = \frac{MN}{PQ}\end{align*}
Now we have been given side lengths for each pair of corresponding sides. These have been written in a proportion or a set of three equal ratios. Remember that there is a relationship between the corresponding sides because they are parts of similar triangles. The side lengths of the similar triangles form a proportion.
Let’s substitute the given measurements in our formula.
\begin{align*}\frac{6}{3} = \frac{8}{4} = \frac{4}{2}\end{align*}
There is a pattern with the ratios of corresponding sides. You can see that the measurement of the each side of the first triangle divided by two is the measure of the corresponding side of the second triangle.
We can use patterns like this to problem solve the length of missing sides of similar triangles.
Let’s look at an example.
Here we have two similar triangles. One is larger than the other, but they are similar. They have the same shape but a different size. Therefore, the corresponding sides are similar.
If you look at the side lengths, you should see that there is one variable. That is the missing side length. We can figure out the missing side length by using proportions. We know that the corresponding side lengths form a proportion. Let’s write ratios that form a proportion and find the pattern to figure out the length of the missing side.
\begin{align*}\frac{AB}{DE} & = \frac{AC}{DF} = \frac{BC}{EF} \\ \frac{5}{10} & = \frac{15}{x} = \frac{10}{20}\end{align*}
Looking at this you can see the pattern. The side lengths of the second triangle are double the length of the corresponding side of the first triangle.
Using this pattern, you can see that the length of \begin{align*}DF\end{align*} in the second triangle will be twice the length of \begin{align*}AC\end{align*}. The length of \begin{align*}AC\end{align*} is 15.
15 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 30
The length of \begin{align*}DF\end{align*} is 30.
Practice solving these proportions.
- \begin{align*}\frac{6}{12} = \frac{x}{24} = \frac{3}{6}\end{align*}
- \begin{align*}\frac{12}{x} = \frac{16}{4} = \frac{20}{5}\end{align*}
- \begin{align*}\frac{8}{2} = \frac{16}{4} = \frac{x}{1}\end{align*}
Check your proportions with a friend to be sure that your answer works. Then draw a pair of triangles to match each set of dimensions and label the side lengths.
V. Use Similar Figures to Measure Indirectly
We can use the properties of similar figures to measure things that are challenging to measure directly. We call this type of measurement indirect measurement.
Let’s look at an example so that we can understand indirect measurement.
Example
Jamie’s Dad is six feet tall. Standing outside, his shadow is eight feet long. A tree is next to him. The tree has a shadow that is sixteen feet long. Given these dimensions, how tall is the tree?
That is a good question! Think about a tree, it makes a shadow and the line from the end of the shadow to the top of the tree creates a triangle. It sounds confusing, but here is a diagram to help you.
Here is a palm tree. You can see from the picture that the tree itself is one side of the triangle, that the shadow length is another side of the triangle and that the diagonal from the top of the tree to the top of the shadow forms the hypotenuse (the longest side) of the triangle.
How would this work with the shadow of a person? Let’s look at an example.
There is a triangle here too. Just because it is on an angle don’t let that fool you. It is still a triangle.
Alright, now let’s go back to the problem again.
Example
Jamie’s dad is six feet tall. Standing outside, his shadow is eight feet long. A tree is next to him. The tree has a shadow that is sixteen feet long. Given these dimensions, how tall is the tree?
To solve this, we have to create two ratios. One will compare the heights of the man and the tree the other will compare the lengths of the shadows. Together, they will form a proportion because similar triangles are proportional and we have already seen how triangles are created with people or things and shadows.
\begin{align*}\frac{Height\ of\ Man}{Height\ of\ Tree} = \frac{Shadow\ length\ of\ man}{Shadow\ length\ of\ tree}\end{align*}
Now we can fill in the given information.
\begin{align*}\frac{6’}{x} = \frac{8’}{16’}\end{align*}
We are looking for the height of the tree, so that is where our variable goes. Now we can solve the proportion.
Our answer is 12 feet. The tree is 12 feet tall.
You can use similar triangles and proportions to measure difficult things. Indirect measurement makes the seemingly impossible, possible!!
Real Life Example Completed
The Pair of Skateboard Ramps
Here is the original problem once again. Reread the problem and underline any important information.
Marc, Isaac and Isabelle thought that designing a skateboard ramp would be easy. Because of this, they have decided to build two of them in their skatepark. Using the computer, they found the measurements for the first skateboard ramp design.
It has the form of a triangle and is in three dimensions, so it also has a width. Here are the dimensions for the first ramp.
28” long \begin{align*}\times\end{align*} 38.5” wide \begin{align*}\times\end{align*} 12” high
Isaac writes the following proportion on a piece of paper.
\begin{align*}\frac{28”}{14”} = \frac{38.5”}{\Box} = \frac{12”}{6”}\end{align*}
“The two ramps are going to be similar, but not congruent,” Isaac begins to explain.
At that moment, his mom begins calling him and he dashes out the door leaving Isabelle and Marc with his work and with the proportion.
“What is the difference between similar and congruent?” Isabelle asks.
“I am not sure,” Marc says. “But he didn’t finish the measurement either. What is the width of the second ramp?”
First, let’s review the difference between similar figures and congruent figures.
A similar figure is one that is the same shape but a different size from the original one. The measurements of similar figures have a relationship. They are proportional. In other words, their dimensions form a proportion.
Congruent figures are the same size and shape exactly. Congruent figures would have the same measurements.
The ramp dimensions are similar. Isaac left Marc and Isabelle with that much information, which means that the dimensions of the ramps are proportional but not exact. Here is the proportion of measurements that Isaac wrote. We can solve the proportion.
\begin{align*}\frac{28”}{14”} = \frac{38.5”}{\Box} = \frac{12”}{6”}\end{align*}
If you look at the completed dimensions, you will see that the length and the height have been divided in half. One ramp will be half the size of the other ramp. Therefore, we can take the width of the ramp, 38.5” and divide it in half to get the width of the second ramp.
38.5 \begin{align*}\div\end{align*} 2 \begin{align*}=\end{align*} 19.25”
This is the width of the second ramp.
Marc and Isabelle have solved the dilemma without Isaac.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Congruent
- having the same size and shape and measurement
- Similar
- having the same shape, but not the same size. Similar shapes are proportional to each other.
- Corresponding
- matching-corresponding sides between two triangles are sides that match up
- Ratio
- a way of comparing two quantities
- Proportion
- a pair of equal ratios.
- Indirect Measurement
- using the characteristics of similar triangles to measure challenging things or distances.
Technology Integration
Khan Academy Congruent and Similar Triangles
James Sousa, Congruent and Similar Triangles
Other Videos:
- http://www.xtremeskater.com/ramp-plans/quarter-pipe/ – Here is a video on how to build a quarter pipe like the one in the introduction problem.
Time to Practice
Directions: Identify the given triangles as visually similar, congruent or neither.
1.
2.
3.
4.
5.
Directions: Use the following triangles to answer the questions.
6. Are these two triangles similar or congruent?
7. How do you know?
8. Side \begin{align*}DE\end{align*} is congruent to which other side?
9. Side \begin{align*}DF\end{align*} is congruent to which other side?
10. Side \begin{align*}EF\end{align*} is congruent to which other side?
11. If the side length of \begin{align*}DE\end{align*} is 10, what is the side length of \begin{align*}GH\end{align*}?
12. If the side length of \begin{align*}HI\end{align*} is 8, which other side is also 8?
13. Are these two triangles similar or congruent?
14. How do you know?
15. Which side is congruent to \begin{align*}AB\end{align*}?
16. Which side is congruent to \begin{align*}AC\end{align*}?
17. Which side is congruent to \begin{align*}RS\end{align*}?
18. Look at the following proportion and solve for missing side length \begin{align*}x\end{align*}.
\begin{align*}\frac{7}{3.5} & = \frac{x}{3.5} = \frac{6}{y} \\ x & = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
19. What is the side length for \begin{align*}y\end{align*}?
20. How did you figure these out?
Directions: Use what you have learned about similar triangles and indirect measurement to solve each of the following problems.
21. If a person who is five feet tall casts a shadow that is 8 feet long, how tall is a building that casts a shadow that is 24 feet long?
22. If a tree stump that is two feet tall casts a shadow that is one foot long, how long is the shadow of a tree that is ten feet at the same time of day?
23. If a 6 foot pole has a shadow that is eight feet long, how tall is a nearby tower that has a shadow that is 16 feet long?
24. If a lifeguard tower is ten feet tall and casts a shadow that is eight feet long, how tall is a person who casts a shadow that is four feet long?
25. Draw the triangle in on the following picture.