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# Triangle Identification as Similar, Congruent, or Neither

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Have you ever built a ramp? Take a look at this dilemma.

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 $\times$ 38.5” wide $\times$ 12” high

Isaac writes the following proportion on a piece of paper.

$\frac{28''}{14''} = \frac{38.5''}{\Box} = \frac{12''}{6''}$

“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.

There are two problems here. One has to do with similar and congruent triangles. The other has to do with the missing measurement.

In this Concept, we will tackle the first problem. The second one regarding the missing measure we will work on in another Concept.

### Guidance

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.

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.

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.

#### Example A

Solution: Neither

#### Example B

Solution: Similar

#### Example C

Solution: Congruent

Now let's go back to the question about the ramps. Here is the original problem once again.

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 $\times$ 38.5” wide $\times$ 12” high

Isaac writes the following proportion on a piece of paper.

$\frac{28''}{14''} = \frac{38.5''}{\Box} = \frac{12''}{6''}$

“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.

Let’s review the difference between similar figures and congruent figures. Solving the proportion will be in another Concept.

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.

### Vocabulary

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.

### Guided Practice

Here is one for you to try on your own.

Are there two triangles similar, congruent or neither?

When you look at these two triangles, you can see that they are exactly alike. The problem could be misleading if you look at the angle measures, but notice that different angle measures have been given.

These two triangles are congruent.

### Practice

Directions: Identify the given triangles as visually similar, congruent or neither.

1.

2.

3.

4.

5.

Directions: Answer each of the following questions.

6. Triangles $ABC$ and $DEF$ are congruent. Does this mean that their angle measures are the same? Why?

7. True or false. If triangles $DEF$ and $GHI$ are similar, then the side lengths are different but the angle measures are the same.

8. True or false. Similar figures have exactly the same size and shape.

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

10. Triangles $LMN$ and $HIJ$ are similar. If this is true, then the side lengths are the same, true or false.

11. True or false. To figure out if two figures are similar, then their side lengths form a proportion.

12. Define similar figures

13. Define congruent figures.

14. Use a ruler to draw a congruent pair of triangles.

15. Use a ruler to draw a pair of triangles that is similar.