Geometry: What Should You Bring to a Formal Proof?
What Should You Bring to a Formal Proof?
A formal proof of a statement is a sequence of steps that links the hypotheses of the statement to the conclusion of the statement using only deductive reasoning. The hypotheses and conclusion are usually stated in general terms. For example, you have already learned that when two lines intersect, the vertical angles formed are congruent. This statement is vague in that it does not name the specific lines that intersect or give the point of intersection. The statement does not even name the vertical angles formed. Because everything is so general, it's hard to get a handle on where to start. That's why in geometry a picture is worth a thousand words. When you draw a picture illustrating what's going on, it's easier to see what you are assuming and what you are trying to prove. You can translate your general statement into the picture in Figure 8.1. Here you have two intersecting lines, ↔AC and ↔BD. They intersect at O. They form a couple of vertical angles, which you've named ∠AOB and ∠DOC. Your mission is to show that these two angles are congruent.
By drawing a picture, you've translated the general statement of the theorem into a specific example that you can pick apart and analyze. Try and include all of your assumptions in your picture. It's helpful to have your picture as complete as possible before beginning your proof, but that's not always possible. Sometimes as you work through a problem you realize that you need to label more things. You can always add labels to your pictures. Your diagram doesn't need to be set in stone until the proof is done.
After you've drawn your picture and labeled the important points, segments, lines, and so on, you have to interpret the hypotheses in terms of your picture. For the vertical angle example, you are given two lines that intersect. In your picture, the two intersecting lines are ↔AC and ↔BD. In mathematical terms, you write that ↔AC intersects ↔BD at O.
The next step is to interpret what you would like to prove in terms of your picture. The general conclusion is that vertical angles are congruent. In your picture, the vertical angles are ∠AOB and ∠DOC. You will prove that ∠AOB ∼= ∠DOC.
A formal proof has a definite style and format consisting of five essential elements.
- Statement. This states the theorem to be proved.
- Drawing. This represents the hypothesis of the theorem. Sometimes you have to translate the statement of the theorem into the specifics of your drawing.
- Given. This interprets the hypothesis of the theorem in terms of your drawing.
- Prove. This interprets the conclusion of the theorem in terms of your drawing.
- Proof. This orders a list of statements and justifications beginning with the given information and ending with what you wanted to prove. There must be a logical flow in this proof.
The last step (and possibly the most difficult of them all) is to start with the given information and prove what you want to prove. This is done by thinking about your definitions, postulates, and any previous theorems that you have established. You want to use deductive reasoning to connect the given information to the conclusion.
Let's practice writing a formal proof by proving Theorem 8.2.
Theorem 8.2: If two lines intersect to form consecutive congruent angles, then these lines are perpendicular.
↔AB and ↔CD intersect at O.
- State the theorem. The theorem you will set out to prove is that if two lines intersect, and the consecutive angles are congruent, then these lines are perpendicular.
- Draw a picture. Figure 8.2 will help us visualize what is going on. We have two lines, ↔AB and ↔CD, that intersect at O. We have four angles to draw: ∠1, ∠2, ∠3, and ∠4. Notice that ∠1 and ∠2 are consecutive angles.
- Given: ↔AB and ↔CD intersect at O, and ∠1 ∼ ∠2.
- Prove: ↔AB ⊥ ↔CD.
- Write the proof.
Before you write out your columns, think about what it means for two lines to be perpendicular. Well, ↔AB ⊥ ↔CD means that the two lines meet to form a right angle. So you need to show that when ↔AB and ↔CD intersect one right angle is formed. It doesn't matter which one is right; if one of them is right, they are all right. You are given that ∠1 ∼= ∠2, and from your drawing you see that ∠1 and ∠2 are supplementary. So a little algebra will show that m∠1 = 90º, and hence is a right angle. So by the definition of perpendicular lines, ↔AB ⊥ ↔CD. When you have your drawing, you will want to look at it to come up with a game plan. Then you can put some columns down and write your proof.
|1.||↔AB intersects ↔CD at O, and ∠1 ∼= ∠2||Given|
|2.||m∠1 = m∠2||Definition of ∼=|
|3.||∠AOB is a straight angle, and m∠AOB = 180º||Definition of straight angle|
|4.||m∠1 + m∠2 = m∠AOB||Angle Addition Postulate|
|5.||m∠1 + m∠2 = 180º||Substitution (steps 3 and 4)|
|6.||m∠1 + m∠1 = 180º||Substitution (steps 2 and 5)|
|7.||2m∠1 = 180º||Algebra|
|8.||m∠1 = 90º||Algebra|
|9.||∠1 is right||Definition of right angle|
|10.||↔AB ⊥ ↔CD||Definition of perpendicular lines|
The first four steps involved in writing a formal proof deal mainly with reading the statement, drawing a picture, and interpreting what is given and what you are trying to prove in terms of your diagram. It's the last step that will require some thought. Before you draw your columns, think through what you are given. Try to imagine which definitions, postulates, or theorems would be useful in connecting the given information with the conclusion. When you have a game plan, everything should fall into place.
Remember to always start your proof with the given information, and end your proof with what you set out to show. As long as you do that, use one reason at a time, and only use definitions, postulates, and other theorems for your reasons, your proofs will flow like a mountain stream.
Put Me in, Coach!
Here's your chance to shine. Remember that I am with you in spirit and have provided the answers to these questions in Answer Key.
- 1. Is the following argument valid?
- a) If an angle is a right angle, then it measures 90º.
- b) ∠A is a right angle.
- Therefore m∠A = 90º.
- 2. Write a formal proof of the following theorem:
- Theorem 8.3: If two angles are complementary to the same angle, then these angles are congruent.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.