Wednesday, October 28, 2020 # Congruent and similar polygons

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A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which third graders have to identify congruent and similar polygons by visual inspection.

Sometimes we have to determine whether a polygon is congruent or similar to another polygon just by looking at the two polygons.

There are different ways that we can compare figures. One way is to count the number of sides. You probably know the names for some of these figures.

#### Triangle: 3 sides  #### Pentagon: 5 sides #### Hexagon: 6 sides #### Congruent figures

We can compare figures that have the same number of sides directly. If we have two triangles, we can check to see if they are congruent.

Congruent is a math word that means exactly the same shape and exactly the same size.

The triangles may be flipped over or moved around, but as long as they are the same shape and size, they are said to be congruent.

One way to check is to cut out one of the triangles and try to place it over the other one. If they match up exactly, they’re congruent.

Sometimes you can’t cut out the figures, so you need to try to match them up in your mind. Another way would be if you have a ruler, you can measure the sides of the shapes and match them up that way.

For example, the two triangles below are congruent, because they are exactly the same shape and size. Yes, one of them is rotated from the other, but if you rotated it back, you would see that one fits exactly on top of the other. That means they’re congruent. #### Similar figures

Some figures look quite a bit alike, but they’re not the same size. They may be similar figures.

Similar figures have the same shape, but the sides may not be the same size.

If we have two triangles, for example, we can check to see if they’re similar. One way to see is to cut out the smaller triangle and then to place it in the center of the other triangle. If there’s an equal amount of space all around the smaller triangle, then the triangles are similar.

Another way to check is to match up the angles of the two triangles. If the angles all match, then the triangles are similar.

The drawing below has two pairs of similar triangles. On the left, we can see that there’s an equal amount of space all around the smaller triangle. That means the two triangles in the drawing on the left are similar.

And in the drawing on the right, we have placed the smaller triangle so that the top angle is right on top of the top angle in the bigger triangle. This shows that all the angles match, so it also shows that the two triangles in the diagram at the right are similar. If you need additional practice on this subject, please visit our online card catalog at VoxLearn.org and search for “congruent” or “similar polygons.”

1. #### Try a few on your own

Look at this pentagon: Which of the following shapes is congruent to the pentagon drawn above? Is the green shape congruent? Well, it looks like it has five points, but the sides of this figure are not exactly straight. They’re all curvy. This one can’t be congruent, because it’s not really the same shape as the original.

What about the yellow one? It looks like the original, but let’s see what happens when we try to put one on top of the other. Remember: they have to line up all the way around in order to be congruent. This one can’t be congruent, either. Even though it’s the same shape, it’s not the same size as the original. The two shapes are similar, because as you see when I put one on top of the other, there’s the same amount of space all the way around the smaller pentagon. But just because they’re similar doesn’t necessarily mean they’re congruent.

The blue one maybe? Well, this one has five vertices like the original and it has five sides like the original. It looks like it’s about the same size, but what about the shape. The original pentagon is a regular pentagon: all its angles are the same, and all its sides have the same length. This one is bent out of shape a little.

Just for fun, look at what happens when I try to put the blue pentagon on top of our original purple pentagon: As you can see, I lined up the top vertex on each pentagon and tried to fit the blue one directly on top of the original purple one. No luck. I can still see part of the purple original below the blue pentagon, so they are not exactly the same shape.

And that means they’re not congruent.

It must be the red one then. Right, but when we first look at it, we can’t really line it up with the original. That’s because it has simply been rotated. Before we can put it on top of the original, we have to rotate it back, as shown by the arrow in the diagram below: After we rotate it so that the top vertex on the red one is in the same place as the top vertex on our purple original, then we can put one directly on top of the other, lining up the top vertex on each. When we do this, we see that the red one is exactly the same shape and size as the purple one, meaning they are congruent.

As you see, it doesn’t really matter about the orientation. If you have to rotate one of the shapes to see whether or not it’s the same shape and size as the other, go right ahead.

2. #### Want to try another one?  Remember: “similar” means they have the same shape but they may have different sizes.

If you said the second one, you were correct. It has the exact same shape as the original, but it’s a bit smaller. Let me draw them right on top of each other. As you see, there’s the same amount of space all the way around the smaller quadrilateral, which looks like it’s completely inside the original quadrilateral.

The first one can’t be correct, because it’s a parallelogram, and a parallelogram (that has two obtuse and two acute angles) doesn’t have the same shape as a square, which is what our original was. When I draw the parallelogram directly on top of the square, you can see that there’s a part of the square sticking out on two of the corners. That is not what it’s supposed to look like for similar shapes. If the shapes were actually similar, we would be able to see an equal amount of one figure under the other all the way around.

And it can’t be the third one, either, since that’s a trapezoid. A trapezoid doesn’t have the same shape as a square, so that can’t be similar. For proof, just draw the trapezoid on top of the square: The top two corners of the square stick out, even though the height is the same and the bottom vertices line up.

Finally, it can’t be the fourth one either. The shape we started with is a perfect square, and drawing that on top of the rectangle (the fourth answer choice) will prove that the shapes are not similar. There is part of the rectangle showing at each of the sides, but there is absolutely no part of the rectangle showing at the top or at the bottom.

In order for these two shapes to be similar, there would have to be an equal amount of space all the way around, and since there is less space on top and bottom than there is on the sides, the two shapes are not similar.

3. #### Illinois Alignment

Illinois Assessment Framework: 9.3.10 (3rd grade) Identify congruent and similar figures by visual inspection.

Illinois Learning Standard 9.B.2 (late elementary): Compare geometric figures and determine their properties including parallel, perpendicular, similar, congruent and line symmetry.