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Common Core Standards: Math

Grade 8

Geometry 8.G.A.2

2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

You know how some identical twins look exactly like each other? Like Mary-Kate and Ashley, Fred and George Weasley, or Lindsay Lohan and…well, Lindsay Lohan?

If George Weasley were to run through a doorway leaving a George-shaped hole, Fred could waltz into this hole and he'd be a perfect fit in every way. That's what's called being congruent. Students should understand congruence as being identical—where all corresponding angles and sides are exactly the same.

Sometimes, it's not easy to tell if two shapes are congruent. If one twin lies down and the other stands upright, they might not look exactly identical. All we have to do to verify that they are congruent is have the both of them either standing up or lying down (which is difficult to do when they're both set on proving how different they are from one another).

Students should know that this happens to shapes, too. Sometimes, we have to pick them up, turn them, flip them, and move them around to see that they are actually congruent. If they are, then the only movements—or transformations—we need to perform are translation, reflection, or rotation.

Being familiar with the three congruence transformations is a good start, but students should know be able to describe which transformations are necessary to see the congruence between two figures as well as identify when two shapes aren't congruent.

Drills

  1. Which of the following is true about two congruent figures?

    Correct Answer:

    They are both the same size and the same shape

    Answer Explanation:

    "Congruent" is geometric lingo for "the same in every possible way." If two shapes are congruent, they're just like identical twins: the only way to tell them apart is using their names. In other words, (D) is right because in order to be equal in every way, they need to be both the same size and shape.


  2. Which of the following is congruent to this figure?

    Correct Answer:

    Answer Explanation:

    For two figures to be congruent, they need to be identical in every way. Being the same size or having the same outline isn't enough. The two have to be a slide, turn, or flip away from being the exact same figure. We're dealing with a shaded in semicircle, and the only one that works is (C).


  3. Which of the following is congruent to this figure?

    Correct Answer:

    Answer Explanation:

    Even though each of the shapes is a triangle, only choice (A) is the same size and shape as the original. Comparing size should be enough to realize that (C) is too tall, while (B) and (D) have significantly different angles from the original triangle. The only option left is (A), which is a reflected version of the given triangle.


  4. Which transformations separate the image from its preimage?

    Correct Answer:

    None of these

    Answer Explanation:

    We can clearly see that the image is not congruent to its preimage. One is a square and one is a rectangle. It's not rocket science. Reflection, translation, and rotation are all congruence transformations and the two aren't congruent, so there has to be some other type of transformation that turns the preimage into the image. That's why the answer is (D).


  5. If you know that ΔABC is congruent to ΔDEF, which of the following is a logical conclusion?

    Correct Answer:

    Both have the same angle measures and side lengths

    Answer Explanation:

    What makes the Weasley twins identical? (We're dealing with math, so let's steer clear of eggs splitting in the womb.) In order for the Weasley twins to be identical as wholes, their parts have to be identical. Being congruent is the same way. Two triangles are congruent not because both of them are right, equilateral, or scalene. They're congruent because they have congruent corresponding parts.


  6. Which type of translation would make the preimage congruent to the image?

    Correct Answer:

    Rotation

    Answer Explanation:

    The difference between our preimage and our image is the fact that it's turned on its side. Translation would keep a figure oriented the same way, so (A) is wrong. Dilation changes the size of the image, so (C) is wrong too. While there is a chance it could be reflection, there's no obvious line of symmetry. The clearest answer is (B).


  7. Which is an example of translation?

    Correct Answer:

    Putting away a folded T-shirt

    Answer Explanation:

    For an object to be translated, it has to change position but not size, shape, or orientation. To figure out which scenario is translation, let's consider the size and shape of the T-shirt in each situation. In (A), the shirt isn't changing location necessarily. In (B) and (C), the shirt is changing size and shape. Only in (D) does the shirt remain the same size and shape, and is moved from one location to another.


  8. Juan has two triangular game pieces that he says are congruent. He holds one behind the other, and the back piece doesn't show. When he reverses the procedure, some of the piece that is now in the back is visible. What does this show?

    Correct Answer:

    The pieces are not congruent

    Answer Explanation:

    At first thought, you may believe that (B) is the right answer. After all, there's the option that the pieces are identical, but one of them is rotated so that they don't line up perfectly. If this were the case, then both pieces would be visible regardless of which direction you look at it. That means the pieces cannot be congruent. We don't have enough information to know whether the pieces are similar to each other or not, so (C) is the right answer.


  9. For which of the following shapes will a translation and a rotation always yield the same image from a preimage?

    Correct Answer:

    Circle

    Answer Explanation:

    Circles look the same from whichever angle we look at 'em. This isn't the case for triangles, rectangles, or even squares. That's why wheels on cars work so well: because they're the same regardless of how they're rotated.


  10. Kyle has a ruler and drawings of two triangles. He claims that simply measuring the lengths of the sides and matching them up will prove that the triangles are congruent. What would you say to him?

    Correct Answer:

    "Correctamundo, Kyle, my man! You couldn't be more right."

    Answer Explanation:

    Triangles have a special quality in that their side lengths essentially determine their angle measures. If we're given three side lengths that make a triangle, there's only one triangle we can make from those lengths. Any two triangles with those side lengths will be congruent.