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Essay Lab | Math Shack | Videos

Common Core Standards: Math

Grade 8

Geometry 8.G.A.5

5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

If you've worked as a babysitter (or ever interacted with a little kid), you've certainly had a conversation very similar to this one at some point in your past:

You: It's time for bed!
Kid: Why?
You: Because it's 10 pm.
Kid: Why?
You: Because I'm the babysitter, and I promised your mom you'd be asleep by now.
Kid: Why?
You: Because little kids need their sleep.
Kid: Why?
You: Because you need sleep to be healthy.
Kid: Why?

And so it goes, long into the night, until you fall asleep while the kid continues to watch TV.

Little kids have an inborn need to have things proven to them. (Need another example? Try telling a toddler that it's a bad idea to put things up her nose. Then get ready to call 911.)

Mathematicians are the same way. They would never accept a rule like, "Never put things up your nose," just because someone says it's true. Even after you tell them it's a bad idea, they'd need to prove it. (Now you know that if mathematicians don't stick pens up their noses, they've proven it to be a bad idea.)

Your students should step up to the plate and prove some geometric rules on their own. Before they do, though, they should know that theorems are proven statements of fact and postulates are logical assumptions that are basic enough not to be proven. They should also understand the properties of lines, angles, and triangles.

Once they've grasped these concepts well enough, they can prove theorems about parallel lines and transversals and their angles. Then, students can take this information and apply it to triangles.

It's important that students know they can apply theorems and concepts from one area of geometry to another. For instance, seeing a triangle as two parallel lines cut by two transversals will help them understand connections between angles in both those situations.

Here's an example of the exterior angle theorem in action that can be shown to students.

 

Drills

  1. Which of the following is probably NOT a good justification for a step in a proof?

    Correct Answer:

    An assumption

    Answer Explanation:

    Assumptions don't carry a whole lot of weight with mathematicians. For an assumption to count, it has to be a postulate. Otherwise, there's no way to verify whether the assumption is logical or not. Theorems are proven and definitions are necessary to know what we're talking about. The only one that doesn't fit is (D).


  2. Which of the following theorems, postulates, or definitions is used in this diagram?

    Correct Answer:

    Alternate interior angles theorem

    Answer Explanation:

    After noticing that the two horizontal lines are parallel, the two remaining sides of the triangle can be thought of as transversals of those parallel lines. All the marked angles are in between the two parallel lines, so we're looking at interior angles rather than exterior angles. If we know the proper name for them, it's clear that these are alternate interior angles, and that alternate interior angles of parallel lines are congruent.


  3. What would this diagram be used to prove?

    Correct Answer:

    Angle sum theorem for triangles

    Answer Explanation:

    To start out, it's important to know what each of the answer options are all about. We can't really prove a postulate (well…we can, but it's not easy), and there are no vertical angles in the diagram. While (C) looks really relevant, we're using it, not proving it. In fact, we're using it to prove (A), because the intersection on the top split three ways proves that all three angles in a triangle add up to 180°, which is the angle sum theorem for triangles.


  4. How could you prove that the sum of the angles of a quadrilateral (that's a 4 sided polygon, like a parallelogram or square) is 360 degrees using the sum of angles in a triangle?

    Correct Answer:

    A quadrilateral can always be split into two 180° triangles

    Answer Explanation:

    Draw any quadrilateral. Then connect 2 alternate vertices that aren't already connected. You'll form 2 triangles, each containing 180°. Since there are two of them, we'll multiply 180° × 2 = 360°. While the ultimate answer in (A) is right, the reasoning isn't correct. We also know that (C) is wrong and (D) isn't true, either.


  5. Using the following diagram, which of the following definitions, theorems, or postulates is NOT applicable?

    Correct Answer:

    Alternate exterior angle theorem

    Answer Explanation:

    We can see that in the diagram, we have three angles that make a triangle and two that form a linear pair (which are interior and exterior angles of the triangle). Knowing that a triangle and linear pair is involved, we'll probably be making use of (A) and (C), so they're out. While (D) has the word "exterior," the theorem actually refers to angles on the exterior of parallel lines, not triangles, so that's our answer. We'd use (B) because angles in a linear pair are supplementary.


  6. If we know that m∠1 + m∠2 + m∠3 = 180 and m∠3 + m∠4 = 180, what can we prove about the diagram below?

    Correct Answer:

    m∠1 + m∠2 = m∠4

    Answer Explanation:

    We can use simple algebra to prove that (C) is the right answer. After rearranging the equations into 180 – m∠3 = m∠1 + m∠2 and 180 – m∠3 = m∠4, we can set the two equal to each other (because both equal 180 – m∠3). This proves that the exterior angle of a triangle is equal to the sum of its two remote interior angles. Say that five times fast.


  7. Which statement is illogical given the fact that the angles of a triangle add up to 180°?

    Correct Answer:

    An obtuse triangle has 3 obtuse angles

    Answer Explanation:

    It's important to know that right angles are 90° in measure, acute angles are less than 90°, and obtuse angles are greater than 90°. In that case, (A) and (B) make sense, and splitting up a pentagon shows that (C) makes sense too. But…how could one triangle have 3 obtuse angles? This would be greater than 270° at the very least. We know that's not true, so (D) is the illogical statement.


  8. Given the diagram below and that m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180, which is a valid conclusion?

    Correct Answer:

    Vertical angles are congruent

    Answer Explanation:

    From what we have, we know m∠1 + m∠2 = m∠2 + m∠3. If we subtract m∠2 from both sides, we end up with m∠1 = m∠3, or ∠1 ≅ ∠3. This proves that vertical angles are congruent. Nowhere is it stated or even implied that any of these three angles have to be right or acute, and while (A) and (D) are correct alone, they have nothing to do with the figure or proof at hand.


  9. Given the figure below and knowledge of parallel lines and vertical angles, which of the following can be proved to be true?

    Correct Answer:

    ∠1 ≅ ∠4

    Answer Explanation:

    Looking at the figure, we can see that ∠1 and ∠3 are congruent because they're vertical angles. We also know that since m and n are parallel, ∠3 and ∠4 are congruent (can you say, "alternate interior angles"?). We've gotten this far, so let's put it all together: ∠1 ≅ ∠3 ≅ ∠4, or more simply, ∠1 ≅ ∠4. There's not enough information to prove any of the other statements.


  10. Given the figure below and knowledge of parallel lines and vertical angles, which of the following can be proved to be true?

    Correct Answer:

    m∠1 + m∠6 + m∠2 = 180

    Answer Explanation:

    We know that a triangle has 180°. Since ∠1 ≅ ∠3, and ∠6 is congruent to the top angle of the triangle. Add in ∠2, and we've got ourselves the three angles of a triangle, which should all add up to 180°. The only way we could prove that the other options are true would be if we had more information (as in m∠1 = 60° or m∠2 = 180 – (m∠3 + m∠1). But we don't.