Wednesday, May 27, 2015

PARCC Practice Test Question 30 (Day 175)

Question 30 of the PARCC Practice Test is on angles. Students are to complete two simple short proofs involving angles:

In the figure shown, CF intersects AD and EH at points B and F, respectively.

Part A:

Given: Angle CBD = BFE
Prove: Angle ABF = BFE

Statements                    Reasons
1. Angle CBD = BFE    1. Given
2. Angle CBD = ABF    2.
3. Angle ABF = BFE    3.

Which two of the given reasons could be used to correctly complete the proof?

(A) Definition of congruent angles
(B) Congruence of angles is reflexive
(C) Congruence of angles is symmetric
(D) Congruence of angles is transitive
(E) Vertical angles are congruent

Part B:
Given: Angle CBD = BFE
Prove: Angle BFE + DBF = 180 degrees

Statements                                      Reasons
1. Angle CBD = BFE                      1. Given
2. Angle CBD + DBF = 180 degrees 2.
3. Angle BFE + DBF = 180 degrees 3.

Which two of the given reasons could be used to correctly complete the proof?

(A) Adjacent angles are congruent
(B) Adjacent angles are supplementary
(C) Linear pairs of angles are supplementary
(D) Reflexive property of equality
(E) Substitution property of equality
(F) Transitive property of equality

This question shouldn't be too difficult to answer. In the proof for Part A, Angles CBD and ABF are vertical angles and therefore congruent, Since CBD is congruent to both BFE and ABF, therefore ABF is congruent to BFE by the Transitive Property. So the correct answers are (D) and (E).

Now in the proof for Part B, this time we have that CBD and DBF form a linear pair, and so these angles are supplementary. Then we can just substitute BFE in for CBD since these angles are given as having equal measures. So the correct answers are (C) and (E).

There are a few issues in this problem, though. Notice that a correct answer for Part A is "Congruence of angles is transitive," whereas most texts write it as, "Transitive Property of Congruence." I once told a student in an Algebra I class where I was student teaching, after she asked which number to multiply first, I told her that multiplication is commutative, instead of use the Commutative Property of Multiplication -- which may have confused her. Then again, notice that Part B calls it "Transitive property of equality."

Then again, one wonders whether we actually use transitivity, whatever we call it. Notice that the Transitive Property usually states that if x = y and y = z, then x = z. But Part A is actually in a different form -- if x = y and x = z, then z = y. Transitivity, in its pure form, requires that the RHS of an equation or congruence be the same as the LHS of another -- not that the LHS of two equations are the same (or the RHS of two equations). An argument can be made that we should rewrite x = z first as z = x, using the Symmetric Property. Now the RHS of one is the LHS of the other, so that we can use the Transitive Property. This is not insignificant, considering that the Symmetric Property is one of the other choices in Part A.

In practice, people call it the Transitive Property even when two LHS or two RHS are the same, even though there's a Symmetric Property. The only time that I've explicitly seen the Symmetric Property in a proof is using the Pappus proof of the Isosceles Triangle Theorem -- in triangle ABC where sides AB and AC are congruent, we are given one S, and using the Symmetric Property, AC = AB is in fact the other S. The A is given by the Reflexive Property as Angle A is congruent to itself, so that Triangles ABC and ACB are congruent by SAS.

But I still haven't addressed the big elephant in the room for this problem. Notice that, believe it or not, the word "parallel" appears nowhere in this problem -- even though the figure associated with this problem looks very much like two parallel lines, AD and EH, cut by a transversal. In both parts, we are given that the corresponding angles are congruent. In Part A, we are asked to show that the alternate interior angles are congruent, and in Part B, we are asked to that the same-side interior angles are supplementary.

Notice that the lines AD and EH don't merely appear to be parallel -- they are provably parallel because we are given that the corresponding angles are congruent. This is what the U of Chicago calls the Corresponding Angles Postulate. But if we were instead given that the lines are parallel, then we would be, in some ways, assuming the converse of the Corresponding Angles Postulate (called the Parallel Lines Postulate in the U of Chicago) and using that postulate to prove two theorems -- one for alternate interior angles and the other for same-side interior angles. So of course we shouldn't use those theorems in the proofs, because we're trying to prove those theorems. Instead, we show how those theorems go back to the Vertical Angle and Linear Pair Theorems.

At any rate, this is a good time to re-evaluate how parallel lines are taught in the U of Chicago and other texts as well as here on the blog, and whether my way is satisfactory or I should find a new way to teach parallel lines, in light of today's featured PARCC question.

Let's begin our review with the U of Chicago text. In many ways, the U of Chicago's presentation of parallel lines is weak. The two postulates mentioned above are in Section 3-4, but neither alternate interior nor same-side interior angles appear until Chapter 5. Moreover, alternate interior angles appear in Section 5-6, but same-side interior angles don't appear at all except in the context of trapezoids in Section 5-5.

Now Dr. Hung-Hsi Wu uses rotations to develop his properties of parallel lines. Theorem 10 states that if lines are parallel then alternate interior and corresponding angles are congruent, and Theorem 12 is the converse of Theorem 10. Notice that rotations lead directly to alternate interior angles, since if we use the diagram for PARCC Question 30, the rotation image of Angle BFE 180 degrees centered at the midpoint of BF is ABF (alternate interior angle), not CBD (corresponding angle). So vertical angles are then used to establish the corresponding angle theorems. Like the U of Chicago, same-side interior angles are just an afterthought for Wu.

We notice that whereas the U of Chicago begins with corresponding angles and Wu begins with alternate interior angles, the Pearson Integrated Math II text that I mentioned last week begins with same-side interior angles! So far, we've seen three different starting points for the teaching of the parallel line theorems.

Dr. Franklin Mason is the one who distinguishes between the Parallel Tests -- statements of the form "if ... then the lines are parallel" and the Parallel Consequences -- "if the lines are parallel then ..." He has altered his parallel lessons several times. Originally Dr. M proved the Triangle Exterior Angle Inequality and used it to prove the Parallel Tests, beginning with the Alternate Interior Angles Test, just like Wu. Then Dr. M made the TEAI a postulate. Finally, he dropped the TEAI Postulate and just made the Corresponding Angles Test a postulate, just like the U of Chicago.

Also, there is the classic Parallel Postulate -- through a point not on a line, there is exactly one line parallel to that line. This is the Parallel Postulate of Playfair. Originally Playfair was the only postulate about parallels mentioned in Dr. M, and he used Playfair to derive the Parallel Consequences but no parallel postulate was needed to derive the Parallel Tests. This is similar to how Euclid proved these theorems many, many centuries ago. To this day Dr. M still uses Playfair only to prove the Consequences -- the Tests go back to the new Corresponding Angles Postulate.

Now which approach best prepares us for the PARCC exam, in particular today's question? Today's featured question shows us how to use corresponding angles to prove the Alternate Interior Angles and Same-Side Interior Angles Consequences. So it's best that we begin with corresponding angles, just like the U of Chicago or the new Dr. M, but not like Pearson, Wu, or the old Dr. M. Again, this is probably why Dr. M switched to making Corresponding Angles a postulate.

On this blog, I started out by using the Wu approach -- which we now consider unacceptable because it leads to Alternate Interior instead of Corresponding Angles. Wu uses Alternate Interior Angles because these naturally flow from his proof based on 180-degree rotations.

I mentioned last summer and fall that what would be more appealing is if instead of rotations, we could use translations to derive the properties of parallel lines. After all, if we take the figure in today's featured PARCC problem and translate Angle BFE by the vector FB, we obtain CBD -- which is indeed the corresponding angle for BFE. And this would justify the use of the name "corresponding angles" for both parallel lines and congruent triangles, since both of them are the preimages and images of certain isometries.

But when we tried this, we ran into a major problem. The properties of translations were proved using parallel lines -- since after all, a translation is the composite of reflections in parallel lines. So we would have circularity if we used translations to prove the parallel line theorems and the parallel line theorems to prove the translation theorems. Instead, I ended up assuming the weakest possible parallel postulate that will allow us to the properties of translations. That turned out to be the Perpendicular to Parallels Theorem, which I took to be our new postulate. This postulate also allowed us to prove Playfair, and then Playfair can be used to derive the Parallel Consequences, just as in Dr. M.

But that's assuming that we've already proved the Parallel Tests. Let's recall how Dr. M proves the Parallel Consequences. He uses a common trick to prove the converses of statements that are much easier to prove -- by combining the forward statement with a uniqueness condition. Thus the Converse of the Pythagorean Theorem is proved using the forward Pythagorean Theorem combined with the uniqueness condition of SSS Congruence (i.e., given three side lengths, there is at most one triangle up to isometry with those lengths). Likewise, Dr. M proves the Corresponding Angles Consequence using the forward statement (the Corresponding Angles Test), with Playfair as the uniqueness condition (through a point and a line, there is at most one line parallel to the given line). This explains why Dr. M chose the Corresponding Angles Test and Playfair as his two postulates.

Now I can only think of two ways to maintain the priority of Corresponding Angles. One is simply to introduce a Corresponding Angles Postulate, as Dr. M does now. The other is to go back to Dr. M's old TEAI method -- a slightly modification of the TEAI proof yields corresponding instead of alternate interior angles. Now that SAS (as well as SSS and ASA) has been moved up to Unit 2, the proof of TEAI itself can now be given.

Notice that we've already given a special case of the Corresponding Angle Postulate here on the blog without any need for TEAI -- the Two Perpendiculars Theorem. We already know that Wu's rotation proofs for parallelograms can be written using triangle congruence instead. It's possible to give his rotation proof of Theorem 10 using congruent triangles as well. If we are given that alternate interior angles such as BFE and ABF above are congruent, we may drop a perpendicular from the midpoint of BF (i.e., the center of the rotation in the original Wu proof) down to line AD. This produces two triangles that are congruent by ASA, and then CPCTC allows us to conclude that newly-drawn perpendicular to line AD must also be perpendicular to line EH. Then the Two Perpendiculars Theorem tells us that AD and EH are parallel lines. But once again, this theorem goes back to alternate interior angles, not corresponding angles.

And so we're left with either TEAI (which can be written to give priority to either corresponding or alternate interior angles) or to use the Wu proof or its ASA modification (which prioritizes alternate interior angles only). Since we want corresponding angles to have priority, we're left with TEAI.

PARCC Practice EOY Exam Question 30
U of Chicago Correspondence: Section 3-2, Types of Angles
Key Theorem: Linear Pair and Vertical Angle Theorems

If two angles form a linear pair, then they are supplementary.
If two angles are vertical angles, then they have equal measures.

Common Core Standard:
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.

Commentary: I give the theorems for linear pairs and vertical angles as these are the only ones needed to answer Parts A and B. But the question hints at the theorems for parallel lines -- and we notice that the Common Core Standard quoted above mentions both the Vertical Angle Theorem itself and the Parallel Line Consequences. Proofs of these are given in Sections 5-5 and 5-6 of the U of Chicago Text. Question 11 of Section 5-6 asks students to prove the Alternate Interior Angles Test.

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