Properties of Congruent and Similar Shapes

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  • 0:07 Comparing Shapes
  • 0:49 Congruent Shapes
  • 2:03 Practice with Congruent Shapes
  • 3:29 Similar Shapes
  • 4:35 Practice with Similar Shapes
  • 6:00 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

In this lesson, we'll look at triangles, rectangles and other shapes that share properties. This includes both congruent and similar shapes. We'll also practice identifying the missing properties of these shapes.

Comparing Shapes

Triangles, rectangles, parallelograms... geometric figures come in all kinds of shapes. This diversity of figures is all around us and is very important. For example, making stop signs octagons and yield signs triangles helps us to differentiate them from a distance.

Figures of the same shape also come in all kinds of sizes. The debit card in your wallet and the billboard on the interstate are both rectangles, but they're definitely not the same size. If they were, you'd either never be able to read that billboard, or your wallet would need to be a really inconvenient size.

When we study figures, comparing their shapes, sizes and angles, we can learn interesting things about them. So, let's get to it!

Congruent Shapes

Sometimes the easiest shapes to compare are those that are identical, or congruent. Congruent shapes are figures with the same size and shape. You could also think of a pair of cars, where each is the same make and model. They're alike in every way. Well, until one gets awesomely tricked out.

For a more geometry-based example of congruency, look at these two rectangles:

These two rectangles are congruent.
congruent rectangles

We can see that both figures have the same lengths and widths. Next, look at these hexagons:

These two hexagons are congruent even though they are not turned the same way.
congruent octagons

They aren't turned the same way, but they are congruent. The sides and angles all match. If we took one, turned it and put it on top of the other, you'd see that they match perfectly. That's what being congruent means.

Sometimes, you'll be given special clues to indicate congruency. Let's look at two congruent triangles:

The symbol between the triangles indicates that the triangles are congruent.
congruent triangles

When two shapes, sides or angles are congruent, we'll use the symbol above. We know angle A is congruent to angle D because of the symbols on the angles. Likewise, angle B is congruent to angle E, and angle C is congruent to angle F. We also have the hash marks on the triangles to indicate that line AB is congruent to line DE, line BC is congruent to line EF and line AC is congruent to line DF.

Practice with Congruent Shapes

When you have congruent shapes, you can identify missing information about one of them. Consider these two triangles:

You can use congruency to determine missing information.
congruent triangles

We know they're congruent, which enables us to figure out angle F and angle D. We just need to figure out how triangle ABC lines up to triangle DEF. If AB is congruent to DE, and AC is congruent to DF, then angle A is going to be congruent to angle D. So, angle D is 55 degrees. We could use the same logic to determine that angle F is 35 degrees. Or, we could just know that the sum of the interior angles of a triangle is 180, and subtract 55 and 90 from 180 to get 35. Either way, we now know all the angles in triangle DEF.

Sometimes you have even less information to work with. But, you can still figure out quite a bit. Consider these triangles:

There is enough information given by this diagram to determine the remaining angles.
congruent triangles

All we're given is the statement that triangle MNO is congruent to triangle PQR. We also know the measures of angles O and Q. This is actually everything we need to know to figure out everything about these two triangles. Use the order of the vertices to guide you. M corresponds to P, N to Q and O to R. So, angle M is congruent to angle P, N to Q and O to R. That means angle R is 50 degrees and angle N is 100 degrees. Since we need the angles to add up to 180, angles M and P must each be 30 degrees.

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