IN THIS CHAPTER

Making a game plan

Starting at the start, working from the end, and meeting in the middle

Making sure your logic holds

Chapter 2 started you off with short proofs and a dozen and a half basic theorems. Here, I go through a single, longer proof in great detail, carefully analyzing each step. Throughout the chapter, I walk you through the entire thought process that goes into solving a proof, reviewing and expanding on the half dozen or so proof strategies from Chapter 2. When you’re working on a proof and you get stuck, this chapter is a good one to come back to for tips on how to get moving again. The proof I’ve created for this chapter isn’t so terribly gnarly; it’s just a bit longer than the ones in Chapter 2. Here it is:

Making a Game Plan

A good way to begin any proof is to make a game plan, or rough outline, of how you’d do the proof. A game plan is your common sense argument of how you’d get from the given facts to the thing you’re trying to prove. The nice thing about a game plan is that you don’t need to worry about what theorems you’ll need to use or how you’ll write out the formal proof.

When you’re working through a game plan, it’s sometimes a good idea to make up arbitrary numbers for the segments and angles in the givens and for unmentioned segments and angles. You should not, however, make up numbers for segments and angles that you’re trying to show are congruent. This optional step makes the proof diagram more concrete and makes it easier for you to get a handle on how the proof works.

Here’s one possible game plan for the proof we’re working on: The givens provide you with two pairs of perpendicular segments; that gives you for and . Then, say congruent angles and are both . That would make and both equal to . Next, because and are complementary, as are and , and would both be . Angles 5 and 8 are congruent vertical angles, as are and , so and would also have to be — and thus they’re congruent. Finally, because , is bisected. That does it.

Using All the Givens

Perhaps you don’t follow the game plan in the previous section — or you get it but don’t think you would’ve been able to come up with it on your own in one shot — and so you’re staring at the proof and just don’t know where to begin. My advice: Check all the givens in the proof and ask yourself why they’d tell you each given.

Every given is a built-in hint. Look at the five givens in this proof. It’s not immediately clear how the third, fourth, and fifth givens can help you, but what about the first two about the perpendicular segments? Why would they tell you this? What do perpendicular lines give you? Right angles, of course. Okay, so you’re on your way — you know the first two lines of the proof (see Figure 3-1).

Using If-Then Logic

Moving from the givens to the final conclusion in a two-column proof is like knocking over a row of dominoes: Just as each domino knocks over the next domino, each proof statement leads to the next statement. The if-then sentence structure of each reason in a two-column proof shows you how each statement “knocks over” the next statement. In Figure 3-1, for example, you see the reason “if two segments are perpendicular, then they form a right angle.” The perpendicular domino (statement 1) knocks over the right-angle domino (statement 2). This process continues throughout the whole proof.

Make sure that the if-then structure of your reasons is correct.

• The idea or ideas in the if clause of a reason must appear in the statement column somewhere above the line of that reason.
• The single idea in the then clause of a reason must be the same idea that’s in the statement directly across from the reason.

Look back at Figure 3-1. Because statement 1 is the only statement above reason 2, it’s the only place you can look for the ideas that go in the if clause of reason 2. So if you begin this proof by putting the two pairs of perpendicular segments in statement 1, then you have to use that information in reason 2, which must therefore begin “if segments are perpendicular, then …”

Now say you didn’t know what to put in statement 2. The if-then structure of reason 2 helps you out. Because reason 2 begins “if two segments are perpendicular …” you’d ask yourself, “Well, what happens when two segments are perpendicular?” The answer, of course, is that right angles are formed. The right-angle idea must therefore go in the then clause of reason 2 and right across from it in statement 2.

Okay, now what? Well, think about reason 3. One way it could begin is with the right angles from statement 2. The if clause of reason 3 might be “if two angles are right angles …” Can you finish that? Of course: If two angles are right angles, then they’re congruent. So that’s it: You’ve got reason 3, and statement 3 must contain the idea from the then clause of reason 3, the congruence of right angles. Figure 3-2 shows you the proof so far.

When writing proofs, you need to spell out every little step as if you had to make the logic clear to a computer. For example, it may seem obvious that if you have two pairs of perpendicular segments, you’ve got congruent right angles, but this simple deduction takes three steps in a two-column proof. You have to go from perpendicular segments to right angles and then to congruent right angles — you can’t jump straight to the congruent right angles. That’s the way computers “think”: A leads to B, B leads to C, and so on.

Chipping Away at the Problem

Face it: You’re going to get stuck at one point or another while working on proofs. Here’s a tip for getting unstuck.

Try something! When doing proofs, you need to be willing to experiment with ideas using trial and error. Doing proofs isn’t as black and white as the math you’ve done before. You often can’t know for sure what’ll work. Just try something, and if it doesn’t work, try something else. Sooner or later, the whole proof should fall into place.

So far in the proof in this chapter, you have the two congruent angles in statement 3, but you can’t make more progress with that idea alone. So check out the givens again. Which of the three unused givens might build on statement 3? There’s no way to know for sure, so you need to trust your instincts, pick a given, and try it.

The third given says . That looks promising because angles 1 and 2 are part of the right angles from statement 3. You should ask yourself, “What would follow if and were, say, ?” You know the right angles are , so if and were , then and would both have to be and thus, obviously, they’d be congruent. That’s it. You’re making progress. You can use that third given in statement 4 and then state that in statement 5.

Figure 3-3 shows the proof up to statement 5. The bubbles and arrows show you how the statements and reasons connect to each other. You can see that the if clause of each reason connects to a statement from above the reason and that the then clause connects to the statement on the same line as the reason. Because I haven’t gone over reason 5 yet, it’s not in the figure. See whether you can figure out reason 5 before reading the explanation that follows.

So, did you figure out reason 5? It’s angle subtraction because and in statement 5 ended up being angles (assuming and were ), and you get the answer of by doing a subtraction problem, . You’re subtracting two angles from two other angles, so you use the four-angle version of angle subtraction (see Chapter 2). Reason 5 is, therefore, “If two congruent angles ( and ) are subtracted from two other congruent angles (the right angles), then the differences ( and ) are congruent.”

At this stage, you may feel a bit disconcerted if you don’t know where these five lines are taking you or whether they’re correct or not. Here’s a tip.

If you’re in the middle of solving a proof and can’t see how to get to the end, remember that taking steps is a good thing. If you’re able to deduce more and more facts and can begin filling in the statement column, you’re very likely on the right path. Don’t worry about the possibility that you’re going the wrong way. (Although such detours do happen from time to time, don’t sweat it. If you hit a dead end, just go back and try a different tack.) Don’t feel like you’ve got to score a touchdown (that is, see how the whole proof fits together). Instead, be content with just making a first down (getting one more statement), then another first down, then another, and so on.

Working Backward

Assume that you’re in the middle of a proof and you can’t see how to get to the finish line from where you are now. No worries — just jump to the end of the proof and work backward.

Okay, so picking up where I left off on this chapter’s proof: You’ve completed five lines of the proof, and you’re up to . Where to now? Going forward from here might be a bit tricky, so work backward. You know that the final line of the proof has to be the prove statement: bisects . Now, if you think about what the final reason has to be or what the second-to-last statement should be, it shouldn’t be too hard to see that you need to have two congruent angles to conclude that a larger angle is bisected. Figure 3-4 shows you what the end of the proof looks like.

Try to continue going backward to the third-to-last statement, the fourth-to-last statement, and so on. (Working backward through a proof always involves some guesswork, but don’t let that stop you.) Why might be congruent to ? Well, you probably don’t have to look too hard to spot the pair of congruent vertical angles and and the other pair, and . Okay, so you want to show that is congruent to , and you know that equals and equals . So if you were to know that and are congruent, you’d be home free.

Now that you’ve worked backward a number of steps, here’s the argument in the forward direction: The proof could end by stating in the fourth-to-last statement that , then in the third-to-last that and (because vertical angles are congruent), and then in the second-to-last that by the Transitive Property (for four angles). Figure 3-5 shows how this all looks written out in the two-column format.

Filling in the Gaps

Working backward from the end of a proof is a great strategy. You can’t always work as far backward as I did in this proof — sometimes you can only get to the second-to-last statement or maybe to the third-to-last. But even if you fill in only one or two statements (in addition to the automatic final statement), those additions can be very helpful. After making the additions, the proof is easier to finish because your new “final” destination (say the third-to-last statement) is fewer steps away from the beginning of the proof and is thus an easier goal to aim for.

Okay, let’s wrap up this proof. All that remains is to bridge the gap between statement and the fourth-to-last statement . There are two givens you haven’t used yet, so they must be the key to finishing the proof.

How can you use the givens about the two pairs of complementary angles? Try the plugging-in-numbers idea again. Use the same numbers you used before, and say that congruent angles and are each . Angle 5 is complementary to , so would have to be . Angle 6 is complementary to , so also ends up being . That does it — and are congruent, and you’ve connected the loose ends. All that’s left is to finish writing out the formal proof.

Writing out the Finished Proof

Sound the trumpets! Here’s the finished proof complete with the flow-of-logic bubbles (see Figure 3-6). (This time, I’ve put in only the arrows that connect to the if clause of each reason. You know that each reason’s then clause must connect to the statement on the same line.) If you understand all the strategies and tips covered in this chapter and you can follow every step of this proof, you should be able to handle any proof they throw at you.