GMAT

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Properties of Shapes Quadrilaterals Parallelograms Trapezoids Polygons How to Identify Similar Triangles Applications of Similar Triangles Properties of Congruent And Similar Shapes Applying Scale Factors to Perimeter Area And Volume of Similar Figures Types of Angles Vertical Corresponding Alternate Interior Others Angles And Triangles Practice Problems The Pythagorean Theorem Practice And Application

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GMAT: Properties of Shapes Rectangles Squares And Rhombuses
Properties of Congruent And Similar Shapes

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.

Properties of Congruent And Similar 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.

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

Practice with Similar Shapes

Let's try practicing with a few similar shapes. Here are two similar rectangles:

Images for practice example 1

Can you figure out x? You just need to set up a simple equation: 3/6 = 7/x. Cross multiply: 3x = 42. x = 14. We did it! Here's a pair of triangles:

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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.

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:

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

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.