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8.13: Similarity

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Remember the girls who were looking at the congruent figures in the Congruent Figures Concept? Well, now they are also going to look at similar figures. Here is the problem.

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? In the last Concept you learned about congruent figures, now you will take what you just learned and learn about similar figures too. At the end of the Concept, you will have a good understanding of both similar and congruent figures.

Guidance

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 90^\circ 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.

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.

List the corresponding angles in the figures below.

Angles G and W 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 H and X correspond to each other. So do angles I and Y and angles J and Z . Now we can name these two quadrilaterals: GHIJ is similar to WXYZ .

As we’ve said, the sides in similar figures are not congruent. They are proportional, however. Proportions have the same ratio. Look at GHIJ and WXYZ again. We can write each pair of sides as a proportion.

\frac{GH}{WX},\frac{HI}{XY},\frac{IJ}{YZ},\frac{GJ}{WZ}

The sides from one figure are on the top, and the proportional sides of the other figure are on the bottom.

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?

OP and RS are the shortest sides in each figure. They are proportional, so we write

\frac{OP}{RS}

Now that we’ve got one pair, let’s do the same for the rest.

\frac{NO}{QR}, \frac{MP}{TS}, \frac{MN}{TQ}

Now let’s use what we have learned to check for similarity between figures.

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 50^{\circ} 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 180^{\circ} for a triangle.

&\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}

The angles in the first triangle are 50^{\circ} , 60^{\circ} , and 70^{\circ} . The angles in the second triangle are 50^{\circ} , 50^{\circ} , and 80^{\circ} . 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

\frac{6}{3}, \frac{6}{3}, \frac{4}{1}

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.

\frac{6}{3} = 2\\\frac{6}{3} = 2\\\frac{4}{1} = 4

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 360^{\circ} . 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 it's your turn to try a few on your own.

Example A

How many right angles are in the first two figures?

Solution: 4 right angles

Example B

What makes the two squares similar if they both have four right angles?

Solution: The side lengths are different.

Example C

In the triangle pair, are the two triangles similar or congruent? why?

Solution: The triangles are similar because the side lengths are different. The angle measures are the same.

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.

Guided Practice

Here is one for you to try on your own.

Are these figures similar, congruent or neither?

These are similar figures. They are the same shape, but not the same size.

Video Review

This is a Khan Academy video on similar triangles

Explore More

Directions: Tell whether the pairs of figures below are congruent, similar, or neither.

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

10. What is a proportion?

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

12. Define similar figures

13. Are the two figures similar or congruent?

14. Define congruent figures.

15. True or false. A proportion is formed by a pair of equal ratios.

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Date Created:

Nov 30, 2012

Last Modified:

Dec 17, 2014
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