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Common Core Standards: Math

High School: Geometry

Congruence HSG-CO.C.9

9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Whether students think you're acute or obtuse, the important thing is that they all know you're right. Not only can you prove it to them, you'll get them to prove it for themselves. Yep, it's time to talk about theorems about lines and angles, which all point to the same end goal: proving congruence.

For many students, proofs are the most grueling part of geometry. It's best to ease them in by reminding them that proofs are just logical statements that flow from one to the next. They aren't a secret alien language, no more so than algebra or trigonometry. Honest.

Students should know that a theorem is different from a postulate because it is a formula or statement deduced from a chain of reasoning, or a series of other proofs already accepted. A postulate, on the other hand, is an assumed truth based on rational geometric principles.

Different students prefer different types of proofs. Some take a liking to paragraph proofs so they can provide a stream-of-consciousness rant and have it be an acceptable answer. Others prefer the two-column proof so that all their arguments are clearly laid out in front of them. Be lenient as to the format of the proof at first, but make sure your students know what the point of a proof actually is and what their goal is in writing one.

Knowing basic information and definitions about lines and angles is crucial, since they'll be the building blocks of your students' arguments. For instance, many theorems concerning lines and angles will make use of the fact that a straight line is 180°. Vertical angles come to mind.

You can let students know that once one proof is worked out, they can use the result in future proofs. Much like Pokémon, proofs are meant to be collected and used when tackling difficult problems. So we can use the vertical angle proof to prove another theorem about alternate interior angles.

Can Pikachu do that? We didn't think so.

Here is recap video of vertical angle theorem.

Drills

  1. There are five ways to prove that lines are parallel. Which of the methods below is not one of these five methods?

    Correct Answer:

    Demonstrate that a pair of interior angles on opposite sides is supplementary

    Answer Explanation:

    This is the one that is not a way to prove two lines are parallel. If a pair of interior angles on the same side is supplementary, then the lines are parallel. Angles that are supplementary on opposite sides would not tell us anything about the two lines. Answers (A), (B), and (D) are all valid ways to prove that two lines are parallel.


  2. What is the definition of complementary angles?

    Correct Answer:

    Two angles that add up to 90°

    Answer Explanation:

    Since definitions are the building blocks of proofs, these sorts of definitions should be engrained into the students' heads. Complementary angles are two angles that add up to 90°. Answer (A) is the definition of supplementary angles, (C) is the definition of congruent angles (who would have thought?), and (D) is the definition of adjacent angles.


  3. Two lines intersect, forming two pairs of vertical angles. One of these angles is 140° in measure. What are the measures of the three remaining angles?

    Correct Answer:

    40°, 40°, 140°

    Answer Explanation:

    The rule to remember is that vertical angles are congruent. Since one of the angles is 140°, we know that its corresponding vertical angle must also be 140°. We also know that the other two vertical angles are equal, and because a full circle totals 360°, they must equal 360° – 2(140°) = 80°. That means they must each be 40°.


  4. If we want to prove that x and y are parallel using the alternate exterior angle theorem, which additional piece of information would we need to include for this particular situation?

    Correct Answer:

    The definition of supplementary angles

    Answer Explanation:

    The alternate exterior angles theorem only applies when we have alternate exterior angles to work with. For that to be true, we need to get that 70° on the other side of y. Since the transversal forms a 180° angle, we can use the definition of supplementary angles to translate that 70° to 180° – 70° = 110°, and then use the alternate exterior angles theorem. Answer (A) is incorrect because parallel postulate deals with two lines and a point and not angles, while (B) and (D) are incorrect because we're looking for exterior angles, not vertical ones or alternate interior ones.


  5. A transversal intersects two parallel lines. Which of the following is not true?

    Correct Answer:

    Corresponding angles are supplementary

    Answer Explanation:

    Answers (A) and (D) are true by virtue of vertical angles. Since (A) is true and we know that a straight line makes an angle of 180°, (C) is true as well. The only one that's not true is (B). Corresponding angles are congruent, not supplementary. This makes sense, since one transversal intersecting both parallel lines should have the same effect on each one. Two angles that correspond to each other should be congruent because the transversal has the same effect on each. More mathematically, we could use combinations of the other answer options to prove that (B) is incorrect.


  6. If AD is a perpendicular bisector of CB, what can we use to prove that AC is congruent to AB?

    Correct Answer:

    SAS postulate

    Answer Explanation:

    We know that ∠ADB is congruent to ∠ADC because they're both 90°. That's the whole point of a perpendicular bisector. We also know that BD and CD are congruent and AD is congruent to itself. Since the perpendicular angles are between the corresponding sides, we can use the SAS postulate to prove that that the two triangles are congruent. If that's true, then the corresponding parts of the triangles are congruent to each other, including AC and AB. There are no relevant vertical angles in the diagram, we don't know that all the sides of the triangles are congruent just yet, and the triangle sum theorem has to do with the angles of triangle equaling 180°.


  7. Which of the following angles cannot be assumed without a specific mark in a figure?

    Correct Answer:

    A 90° angle

    Answer Explanation:

    A 180° angle is the same as a straight line. Hopefully we can assume a straight line simply by looking at an image. Otherwise, we'd need to identify every line as having a 180° angle. Since a pair of vertical angles is also formed by the intersection of two lines, it's not necessary to label them in a figure. They can be assumed. A right angle of 90°, however, must be indicated in the figure by a little box in the corner of the angle. The only reason it's not (D) is because it'd be inconvenient to have to identify every line by its 180° angle and every set of vertical angles.


  8. What is the supplement of the complement of 30°?

    Correct Answer:

    120°

    Answer Explanation:

    We have to work backwards here, but don't worry. We'll moonwalk you through it. We'll start with 30°. The complement of 30° is the number that, when added to 30, gives us 90. In other words, 90 – 30 = 60°. Now, we're looking for the supplement of that. The supplement of 60° is the number that, when added to 60, gives us 180. In other words, 180 – 60 = 120°.


  9. Which of the following is always true about an angle bisector?

    Correct Answer:

    It splits an angle into two congruent angles

    Answer Explanation:

    A bisector does exactly what you'd expect it to do: split something into two pieces. So it stands to reason that an angle bisector splits an angle in half. That means the two resulting angles have the same measure, which means they're congruent. According to the definitions of complementary and supplementary angles, (A) is only true when both resulting angles are 45°, and (B) is only true when both resulting angles are 90°. Since there's no mention of parallel lines or transversals, corresponding angles don't apply in this situation. The two angles formed by an angle bisector are congruent to each other.


  10. Which of the following is not necessarily true about perpendicular line segments?

    Correct Answer:

    One line is a perpendicular bisector of the other

    Answer Explanation:

    If we picture two perpendicular line segments, we can see that four 90° angles surround the intersection point. Just like any pair of intersecting lines, they form two pairs of vertical angles, so (A) is true. Since a 90° angle is a right angle, (B) is true as well. A line is essentially a 180° angle, so another line that splits it into two congruent angles is an angle bisector. There is no guarantee, however, that the lines intersect at the midpoint of each line segment. While the angles are split in half, we don't know that the segments are split in half, too. That makes (D) the right answer.


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