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Common Core Standards: Math

High School: Geometry

Congruence HSG-CO.C.10

10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

If your students have gotten tired of lines and angles, they're in luck. Tell them that triangles are nothing like lines and angles. Instead, tell them triangles are new, edgy, and they've got some good points…three of them, actually. Conveniently leave out the fact that triangles are just three line segments joined together.

Students should know the basic definitions that come with triangles and how to classify them based on angles and sides. When they can use the words "equiangular" and "isosceles" in everyday conversation, you'll know you've done a good job.

Students should also be comfortable with the angles of a triangle, both interior and exterior. They should know that all the interior angles of a triangle add up to 180°, and they should know how to prove it. It's better to introduce these concepts to them by using concepts they should already know, like parallel lines and transversals.

But that's just the tip of the triangular iceberg. There's way more inside triangles than just three interior angles. For instance, we can fill a triangle with medians, line segments that join the vertices of a triangle to the midpoints of its opposite sides. We can also connect the midpoints of each side in the triangle to form a similar triangle that's half the size of the original one.

Finally, students shouldn't get lost with all the theorems and postulates. They all build on each other, and it's best to keep track of these proofs and postulates so that students don't get confused. Also, students should know that using proofs and theorems they've already learned isn't cheating; it's applying the skills they've learned, and it's highly encouraged.

If studying triangles still seems more confusing than the Bermuda Triangle, give plenty of examples and draw on knowledge that's already been introduced. It's difficult to learn something new without understanding the basics, so go back and re-derive some proofs if needed. You'll get grumbles in the moment, but they'll thank you for it later. Better than being stuck in the Bermuda Triangle, anyway.

Drills

  1. What is the difference between an equilateral triangle and an equiangular triangle?

    Correct Answer:

    An equilateral triangle has three congruent sides and an equiangular triangle has three congruent angles

    Answer Explanation:

    Equiangular and equilateral can both be used to describe the same triangle: one that has three congruent sides and three congruent angles. While all equiangular triangles are equilateral, and all equilateral triangles are equiangular, the two words are not equal to each other. One refers to the angles and the other refers to the sides. It might be tempting to choose (D), but answers (A) and (B) can't both be true because one negates the other. The only right answer, then, is (B).


  2. Which of the following is unnecessary to prove that the 2 base angles of an isosceles triangle are congruent?

    Correct Answer:

    The angle sum theorem for triangles

    Answer Explanation:

    Starting with an isosceles triangle, we can bisect the top angle and the base with a line segment. In order to do that, we have to use (C). Since the bisector is congruent to itself, the two angles are congruent, and the sides of the triangle are congruent, we can use (B) to prove that the two individual triangles are congruent. Then, (D) says that if the triangles are congruent, then so are the corresponding angles within each triangle, which proves our theorem. The only answer choice we didn't use was (A).


  3. Which of the following is not true about the medians of a triangle?

    Correct Answer:

    A triangle can have three or fewer medians

    Answer Explanation:

    A triangle always has three medians because it has three vertices. All the other stuff is completely true. In fact, they're real conversation starters. Show off your geometry smarts to your friends… or better yet, to that girl or guy you've had your eye on this semester.


  4. Similar triangles are triangles that are the same shape, but not the same size. How can we determine whether triangles are similar or not?

    Correct Answer:

    Both (A) and (C) are correct

    Answer Explanation:

    The important aspect of similar triangles is that they don't need to be the same size, only the same shape. If two triangles can lie directly on top of one another, they are congruent, not similar. However, to ensure that a triangle is similar to another, their angles should be equal and their corresponding sides should all have the same ratios.


  5. The SSS and SAS postulates can be applied to determine whether triangles are similar or not. The angles have to be the same, but the corresponding sides only have to be proportional. Which of the following is another postulate that proves similarity?

    Correct Answer:

    AA postulate

    Answer Explanation:

    The Angle-Angle Postulate, or AA, says that when two angles of a triangle are congruent to two corresponding angles of another, the two triangles are similar. Answer (A) is unnecessary because we only need two angles of a triangle to define the third. Answer (C) is incorrect because if triangles meet this criterion, they're congruent, not similar. The SSA postulate does not exist because two unique triangles can be created when two adjacent sides and an unincluded angle are known, which means no similarity can be proved.


  6. Which of the following does the angle sum theorem for triangles state?

    Correct Answer:

    The three angles of any triangle always add up to 180°

    Answer Explanation:

    If we look at a triangle as two parallel lines and two transversals, we can use several theorems about angles and lines (the definition of supplementary angles, the alternate interior angles theorem) to prove that the three interior angles of any triangle will always add up to 180°. Of course, that means the rest of the answers are incorrect.


  7. A segment connects the midpoints on two sides of a triangle. What is true about this segment?

    Correct Answer:

    It is always parallel to the third side and half as long as the third side

    Answer Explanation:

    Answer (A) is incorrect because whether the line is vertical or horizontal depends on the positioning of the triangle itself and the sides of a triangle aren't always the same length. Answer (B) is incorrect since the segment could not possibly be perpendicular, given the shape of a triangle. It should be clear that the segment is parallel to the third side, but the length of the segment doesn't depend on the length of the two other sides. Therefore, (C) is the right answer.


  8. If two angles of one triangle are congruent to two angles of another triangle, then the remaining angle in one triangle is congruent to the remaining angle in the other. Which of the following makes this statement true?

    Correct Answer:

    The angle sum theorem for triangles

    Answer Explanation:

    The definition of supplementary angles applies to two angles that add up to 180°, and isn't all that relevant in this case. Answer (C) is also irrelevant because it has to do with the sides of a triangle rather than its angles. We're out to prove the congruence of angles, not triangles, so (D) isn't right either. Knowing that all the angles in a triangle add up to 180° allows us to figure out the third angle when we know two. That's thanks to (B).


  9. Which of the following is true about right triangles?

    Correct Answer:

    The acute angles of a right triangle are complementary

    Answer Explanation:

    If we take a look at each option and assume it's true, what would happen? Answer (A) states that the two angles that aren't 90° in a right triangle add up to 90°. Since all the angles in a triangle add up to 180°, this is true. If we have two right angles in a triangle, the other angle would be 180° – 90° – 90° = 0°, or nonexistent. That can't be right. A triangle can have angles greater than 90°; it's called an obtuse triangle. If the hypotenuse were greater than the lengths of the other two sides, the sides wouldn't reach each other and be able to "close." The only option that makes sense is (A).


  10. Which of the following is necessary to prove that a triangle's exterior angle equals the sum of the two remote interior angles?

    Correct Answer:

    The definition of supplementary angles

    Answer Explanation:

    In order to prove that a triangle's exterior angle is equal to the sum of its two remote interior angles, we can use the angle sum theorem for triangles and the definition of supplementary angles, both of which equal 180°. Then, we can set these sums of angles equal to each other and subtract the interior angle we don't need. Only (C) is used in this process.