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Common Core Standards: Math

High School: Geometry

Congruence HSG-CO.B.7

7. Use the definitions of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Students should know that two triangles are congruent if there is a rigid motion that maps one onto the other. All corresponding pairs of all sides and all angles must be congruent, which means the two triangles are essentially equal. The only triangles that don't work like this are love triangles; each one is different in its own heartbreaking way.

Of course, if you've read through the standards before this, you know that rigid motions and congruence aren't limited to triangles. So why do triangles get their own standard? What makes them so special?

Since triangles are defined by 3 sides and 3 angles, knowing a limited number of each is often enough to find the rest of the missing information. In short, congruence is easier to prove with triangles. In fact, triangles have their own postulates that are designed to prove triangle congruence with limited information specifically. Fancy shmancy.

So if all corresponding sides and angles of two triangles are congruent, the two triangles themselves are congruent. In applying the rules of congruence and rigid motions to triangles, students drill down one level deeper: not only are the triangles themselves congruent, but the corresponding parts are congruent, as well. This opens up a whole new world of how to discover more about the use of triangles in the real world.

 

Drills

  1. A gymnast on the parallel bars holds his legs parallel his torso perpendicular to the floor. He then performs a move in which he starts to lean forward, letting his legs move at the same pace as his head and torso. He ends up with his upper body parallel to the floor and his legs perpendicular. If the gymnast's torso and legs make up the legs of a right triangle, are the initial and final triangles congruent?

    Correct Answer:

    Yes, because no lengths or angles changed from the first position to the second

    Answer Explanation:

    Answer (A) is incorrect because position does not matter in determining congruence. Answer (B) is incorrect because the two positions are clearly different. Answer (C) is incorrect because it claims that reflected images are not congruent to one another when, in fact, they can be. That leaves (D) as the answer. If the gymnast kept the angles and distances of his torso and legs constant, the two triangles should be congruent because all individual parts that make up the triangles did not change.


  2. A ladder leans against a wall, forming a right triangle. If the top of the ladder slides down a little from its original resting place, is the new triangle congruent to the one before?

    Correct Answer:

    No, because the angle formed by the ladder changed

    Answer Explanation:

    We can imagine the ladder as the hypotenuse of a triangle, the wall as the vertical leg, and the floor as the horizontal leg. If the ladder slides down a bit, only one part of the entire triangle changes. While the length of the ladder stays the same, the angles it forms with the wall and the floor change. The resulting triangle is no longer congruent because of these angle differences, making (C) the right answer.


  3. Two ladders stand the same distance away and lean up against a wall. One ladder is 3 feet longer than the other. If this ladder is reduced to the same length as the second ladder, which of the following is true?

    Correct Answer:

    The two triangles are congruent and a non-rigid motion was performed.

    Answer Explanation:

    Reducing the length of one of the sides of the triangle transforms the overall triangle in a non-rigid way. However, since the ladders are both placed the same distance away from the wall and are transformed to have the same length, the two final triangles are congruent. Since the triangles are congruent without a rigid motion having been performed, the right answer is (C).


  4. Two slices of watermelon are cut so that they have the same exact interior angles. Are the watermelon slices congruent?

    Correct Answer:

    No, because the triangles have the same interior angles, but their side lengths may be different

    Answer Explanation:

    For two triangles to be congruent, all corresponding sides and all corresponding angles must be congruent. Although all corresponding angles are congruent, we don't know whether or not all the corresponding sides are congruent. It could be that one slice of watermelon is the size of a guinea pig, while the other is the size of Papua New Guinea. Their angles can be the same, but their side lengths might be dramatically different.


  5. We know that for two triangles to be congruent, their side lengths and interior angles must be identical, but there are two other elements of their size that must also be equal to make them truly congruent. What are these two elements?

    Correct Answer:

    Area and perimeter

    Answer Explanation:

    The distances among the three sides of a triangle make up the triangle determine area and perimeter. So if the side lengths are the same between two triangles, then the area and perimeter will be the same as well. Answer (B) is incorrect since a bisector can come from any of the three angles within the triangle and end up at ay point on the opposite side without affecting the triangle's dimensions. A triangle is a two-dimensional figure, so volume is not applicable. A transformation isn't an element of size and not all triangles have hypotenuses. The only answer that even remotely makes sense is (A).


  6. A triangle is reflected, translated, and then rotated. If it's called ΔABC, which of the following is congruent to it?

    Correct Answer:

    ΔA'B'C'

    Answer Explanation:

    When naming two congruent triangles, the order of the points is important. If a triangle is congruent to ΔABC, then the points of the congruent triangle should be listed in the same order as those congruent with A, B, and C respectively. All the transformations performed are rigid, so the two triangles are for sure congruent. In the new triangle, we assume that A' represents A after all the transformations. So ΔABC is congruent to ΔA'B'C'.


  7. Which of the following is true for two congruent triangles ΔABC and ΔDEF?

    Correct Answer:

    B is congruent to ∠E

    Answer Explanation:

    Two congruent triangles are only congruent if all their corresponding angles and sides are congruent. Answer (A) isn't right because the word "transformation" is too general. Only rigid transformations can map one onto the other, but translations and rotations can work just fine, too. That means (D) isn't right, either. Since AB doesn't correspond to EF, this isn't necessarily true for the two triangles. On the other hand, ∠B and ∠E are corresponding angles, so they must be congruent.


  8. Which of the following is true for two non-congruent triangles ΔABC and ΔDEF?

    Correct Answer:

    A transformation can map one onto the other

    Answer Explanation:

    When two triangles are non-congruent, it means that no rigid transformation can map one onto the other, regardless of which one. Other transformations, though, like dilations, compressions, and stretches, are fair game. By process of elimination, we get (B) as our answer.


  9. ΔABC is an isosceles triangle, which means that AB and BC are congruent. If ΔAXY is also isosceles, with XB and YC congruent, which triangles, if any, are congruent?

    Correct Answer:

    ΔABX and ΔACY

    Answer Explanation:

    We know this because all the corresponding sides of ΔABX and ΔACY are congruent to each other. ΔABC is an isosceles triangle, which means AB and AC are congruent. We also know that AX and AY are congruent because ΔAXY is isosceles, too. We're given that XB and YC are congruent, which completes the two triangles. That's all we need to prove that ΔABX and ΔACY are congruent.


  10. Which of the following triangles has an order of rotational symmetry greater than 1?

    Correct Answer:

    Equilateral, a triangle with all angles at 60°

    Answer Explanation:

    When all the angles are equal, the triangle can be rotated in any direction and it will look the same. It's also worth noting that an equilateral triangle has 3 congruent sides as well as 3 congruent angles. None of the other triangles have an order of rotational symmetry greater than 1 because the angles and sides are not congruent to each other. An isosceles triangle translates to an axis of symmetry, but not an order of rotational symmetry.


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