## Tuesday, June 7, 2016

### PARCC Practice Test Question 34 (Day 177)

Question 34 of the PARCC Practice Exam is all about a proof:

34. An incomplete proof of the theorem that the sum of the interior angles of a triangle is 180 degrees is shown.

[In triangle ABC, angles A, B, and C are also labeled 1, 2, and 3 respectively. Auxiliary line BD is drawn in with angle 4 adjacent to side BC and angle 5 adjacent to side BA -- dw]

Given: Triangle ABC
Prove: Angle 1 + Angle 2 + Angle 3 = 180

Proof:
Statements                                                       Reasons
1) Draw line BD parallel to line AC.               1)
2)                                                                      2)
3) Angle 2 + 4 = ABD, Angle 5 + ABD = 180 3) Angle addition postulate
4) Angle 5 + Angle 2 + Angle 4 = 180            4) Substitution property of equality
5) Angle 1 + Angle 2 + Angle 3 = 180            5)

Part A
What is the appropriate reason for the statement in step 1?
A. Through any two points, there is exactly one line.
B. Through a point not on a line, there is exactly one line parallel to the given line.
C. If two lines cut by a transversal form congruent corresponding angles, then the lines are parallel.
D. If two lines cut by a transversal form congruent alternate interior angles, then the lines are parallel.

Part B
Which pairs of angle congruences or equalities should be used for the statement in step 2?
Indicate all such pairs.
A. Angle 1 = Angle 2
B. Angle 1 = Angle 3
C. Angle 1 = Angle 4
D. Angle 1 = Angle 5
E. Angle 2 = Angle 3
F. Angle 2 = Angle 4
G. Angle 2 = Angle 5
H. Angle 3 = Angle 4

Part C
Select from the drop-down menu to correctly complete the sentence.
The reason for the statement in step 2 is that Choose...
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
If two lines cut by a transversal form congruent corresponding angles, then the lines are parallel.
If two lines cut by a transversal form congruent alternate interior angles, then the lines are parallel.

Part D
Select from the drop-down menu to correctly complete the sentence.
The appropriate reason for the statement in step 5 is the Choose...
Reflexive property of equality
symmetric property of equality
transitive property of equality
substitution property of equality

I've discussed the proof of the Triangle-Sum Theorem here on the blog back in November. This is the completed proof that the PARCC expects students to provide:

Proof:
Statements                                                       Reasons
1) Draw line BD parallel to line AC.               1) Playfair's Parallel Axiom (B)
2) Angle 1 = Angle 5, Angle 3 = Angle 4 (DH) 2) Alternate Interior Angles Consequence (A)
3) Angle 2 + 4 = ABD, Angle 5 + ABD = 180 3) Angle addition postulate
4) Angle 5 + Angle 2 + Angle 4 = 180            4) Substitution property of equality
5) Angle 1 + Angle 2 + Angle 3 = 180            5) Substitution property of equality (D)

Here are a few things I want to say about this proof. First of all, we notice that in Part B, the PARCC distinguishes between congruence and equality. Many Geometry teachers just ignore the difference and not make a big deal about it (like my own teacher two decades ago), while some other teachers enforce the distinction. In general, numbers are equal, but geometric figures are congruent. This is why when we use the equal symbol with angles, we must use the measure symbol (m), since it's the measures (numbers) that are equal, not the angles (figures).

(Actually, it would be correct to say that Angle 1 actually equals Angle A, since these literally refer to the same angle. But Angles 1 and 5 aren't equal -- they're merely congruent.)

In the U of Chicago text, congruence of figures is defined by the existence of an isometry mapping one to the other. Therefore we don't know that angles with the same measure are congruent unless we can show the existence of such an isometry. Fortunately, in Lesson 6-5, the Segment Congruence and Angle Congruence Theorems tell us that two segments or angles with the same measure are congruent, so we can use "angles with the same measure" and "congruent angles" interchangeably. I point out that the U of Chicago is careful to avoid using the word "congruent" before Lesson 6-5 -- so, for example, the Vertical Angle Theorem of Lesson 3-2 asserts that vertical angles have equal measures, not that vertical angles are congruent.

Here on the blog, I often use congruence and equality interchangeably despite the Common Core definition of congruence in terms of isometry. This is due to the limitations of ASCII. I once wrote that it's possible to underline a slash to create an angle symbol and write:

m/1 = m/5

But then it's not easy to write the corresponding congruence statement. The best we can do is place a tilde right next to the equal sign:

/1 ~= /5

I chose not to do this because then the equal sign is easy to miss, and then a statement that two triangles are congruent may be mistaken for one that they are merely similar. So I avoid this problem altogether simply by writing "Angle 1 = Angle 5" for both the congruence of the angles and the equality of their measures.

Back in November, I wrote about two more proofs of the Triangle-Sum Theorem -- one that comes from Lesson 5-7 of the U of Chicago text, and one that appeared in a job interview for a teaching position (not the position that I was eventually hired for) and which I included on a worksheet. This is what I wrote about the two proofs seven months ago (for these proofs, I am renumbering the angles to match the PARCC labels in order to avoid confusion):

Here's the proof of the Triangle-Sum Theorem given in the U of Chicago text. Since the book gives a two-column proof, I'll convert it to a paragraph proof:

Triangle-Sum Theorem:
The sum of the measures of the angles of a triangle is 180 degrees.

Given: Triangle ABC
Prove: angle A + angle B + angle C = 180

Proof:
Draw line BD with the measure of angle 5 equal to angle A. By the Alternate Interior Angles Test, lines BD and AC are parallel. Then angle 4 has the same measure as angle C, by the Alternate Interior Angles Consequence. By the Angle Addition Postulate, angles 5, 2 (ABC), and 4 add up to 180 degrees. Substituting, we get that angles AABC, and C add up to 180 degrees. QED

Right now, I am a substitute teacher, but last year I interviewed for a position as a regular teacher, and one of the things I was asked to prove was the Triangle-Sum Theorem. (I also had to derive the Quadratic Formula.) I gave a two-column proof similar to the one given in the text, and the principal told me that it was satisfactory, but that he might have preferred something like this:

Statements                                              Reasons
1. Draw line BD parallel to line AC        1. Uniqueness of Parallels (Playfair)
2. angle 5 = angle A, angle 4 = angle C    2. Alternate Interior Angles Consequence
3. angle 2 = angle ABC                            3. Reflexive Property of Equality
4. angle 5 + angle 2 + angle 4 = 180        4. Angle Addition Postulate
5. angle A + angle ABC + angle C = 180  5. Substitution (steps 2 and 3 into step 4)

So we include step 3, to show students that we are making three substitutions. Calling the same angle by two different names -- angle 2 and angle ABC -- emphasizes the need for a Reflexive Property to show that this angle equals itself. The U of Chicago just changes angle 2 to angle ABC without any explanation whatsoever. On the other hand, the U of Chicago distinguishes between the Angle Addition Postulate and the Linear Pair Theorem (which is the just the Angle Addition Postulate in the case that the angles add up to 180). The hope is that this form of the proof is the best for students to understand, which is our goal.

Today I notice how the U of Chicago and worksheet proofs compare to the PARCC proof. One thing that the U of Chicago and PARCC proofs have in common with each other against my worksheet proof is their handling of the Angle Addition Postulate. Strictly speaking, Angle Addition only tells us when the sum of two small angles is a large angle -- it doesn't permit us to add three or more adjacent angles together. To add three angles, we apply Angle Addition twice and use Substitution to combine the two equations. (The U of Chicago mentions a "Linear Pair Theorem" as a special case of Angle Addition, but the PARCC doesn't.) My worksheet, however, hand-waves over the restriction that only two angles may be added under the Angle Addition Postulate.

Meanwhile, my worksheet and PARCC proofs have something else in common -- both of them draw the auxiliary line BD parallel to AC, and give Playfair as justification. We know that either Playfair or some other statement equivalent to Euclid's Fifth Postulate is necessary in the proof since the sum of the angles of a triangle is 180 only in Euclidean geometry. In the U of Chicago text, the proof begins with the AIA Test, which really is neutral. So that proof indeed requires only one non-neutral line.

On the other hand, both the AIA Test and "at least one line parallel" fail in spherical geometry. It may seem as if I'm making a big deal about nothing, but when we return to spherical geometry this summer, it may be interesting to note which step of the Triangle-Sum Theorem fails in spherical geometry and which step fails in hyperbolic geometry, so that the proof is truly Euclidean. We now see that Step 1 fails in spherical geometry and Step 2 fails in hyperbolic geometry.

So let's answer the most important question -- which version of the Triangle-Sum proof will students understand the most? Here's a link to Cut the Knot about the Triangle-Sum Theorem:

http://www.cut-the-knot.org/triangle/pythpar/AnglesInTriangle.shtml

There are four proofs of the Triangle-Sum Theorem given here. Notice that the proof that we usually see in Geometry classes is the second proof, which is attributed to Pythagoras and his followers, not to Euclid. This proof is simpler than either of our earlier proofs -- there's no need for an extra substitution step to replace an angle with itself or to use Angle Addition with three angles.

Euclid's proof, which is given first, is similar to the the Pythagorean proof except that there is one Alternate Interior Angle Consequence step and one Corresponding Angle Consequence step, as opposed to both AIA steps. But notice that even Euclid himself starts out by drawing a parallel auxiliary line using his Proposition 31 (which is essentially Playfair), rather than use his neutral Proposition 27 to draw the auxiliary line (as the U of Chicago text does).

The third Cut the Knot proof may sound appealing in a Common Core class, as it appears to based mostly on rotations. But this proof turns out to be invalid -- more precisely, its author Thibaut was hoping that the proof would be completely neutral (and thus serve to prove the Fifth Postulate), but it is not. Cut the Knot explains that Thibaut assumed that the composite of rotations is always a rotation, but there's a special case where this composite is a translation instead.

The fourth proof is based on paper folding -- which really means reflections. Just like the rotation proof, its dependence on a Parallel Postulate is rather subtle. All things being said, the third and fourth proofs may appear in a Common Core class, but the second proof will be the easiest for the students to understand.

PARCC Practice EOY Question 34
U of Chicago Correspondence: Lesson 5-7, Sums of Angle Measures in Polygons

Key Theorem: Triangle-Sum Theorem

The sum of the measures of the angles of a triangle is 180 degrees.

Common Core Standard:
CCSS.MATH.CONTENT.HSG.CO.C.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.

Commentary: I could have simply squeezed the entire proof onto one page and asked students to fill in the blanks without multiple choice, but I decided to use both pages and preserve the choices so that the students know what to expect on the computer-based tests. The focus is now pivoting from the PARCC to the final exam, which I will post here tomorrow. So these problems can also serve as preparation for the final.