Geometry proofs

Welcome to the third lesson of this geometry course! (Back to the course main page)

Conditionals

If it is sunny, then I can play outside.
If there are clouds, then it will rain soon.
If you add sodium to water, then you will create an explosion.
If a statement contains if and then, then it is called a conditional.
A conditional is simply an if/then statement. If you accept the part after the "if", also called the hypothesis, then you must accept the statement after the "then", also known as the conclusion. So, all conditionals are in the form
If hypothesis, then conclusion
You may recall using these statements in science class with the scientific method. However, remember that only the statement after the "if" is called the hypothesis. The whole statement is just called the conditional.
Remember, if you accept the hypothesis of a conditional to be true, then the conclusion must also be true.
Now, take a look at some of the old postulates. You'll notice that they can be reworded into conditionals. For example, the postulate which says Through any two points there is only one line can be read as If there are two points, then there is a unique line through the points. Yes, that is awkward wording, but it shows that postulates can be written as conditionals.
If there are three colinear points A, B, and C, and B is between A and C, then AB+BC=AC.
If there are three points, then there is at least one plane through all three points.
If there is a line, then there are at least two points on that line.
There are also some new conditionals which we will introduce in 3, 2, 1...

Conditionals which have traveled from algebra to geometry

The following conditionals are always true. It will make sense if, while you read them, you shout, "These are an insult to my intelligence!" for in fact, that's what they'll seem like.
The first one is called the reflexive property.
If a is a number, then a=a.
The second one is called the symmetric property.
If a=b, then b=a.
The third one is called the transitive property.
If a=b and b=c, then a=c.
Now, explanations of the conditionals. The first one says any number equals itself. Strangely enough, this incredibly obvious statement will be a common sight when you write proofs.
The second one says that equations can be reversed. In other words the statement x=4 is equivalent to the statement 4=x.
The third statement says that if two numbers are equal to the same number, then they are equal. Thus, if x=4 and y=4, then x=y.
The next four conditionals are very similar to each other.
The first one is called the addition property of equality.
If a=b and c=d, then a+c=b+d.
The second one is called the subtraction property of equality.
If a=b and c=d, then a-c=b-d.
The third one is called the multiplication property of equality. (Are you seeing the pattern?)
If a=b and c=d, then ac=bd.
The fourth one is called the division property of equality.
If a=b and c=d and neither c nor d equal 0, then a/c=b/d.
Now, all of these properties are very similar, as you probably can tell; for example, if x=2 and y=3, then
• x+y=2+3 by the addition property of equality.
• x-y=2-3 by the subtraction property of equality.
• xy=2(3) by the multiplication property of equality.
• x/y=2/3 by the division property of equality
You may notice that the division property of equality has an extra part in its hypothesis that isn't in any other property of equality. (Remember the name hypothesis? If hypothesis, then conclusion. If the hypothesis is accepted as true, then the conclusion must be accepted as true as well.) That is because division is undefined when the denominator is 0. 1/0 doesn't exist! Neither does 4/0, -200/0, or 0/0. The extra part of the hypothesis is to prevent division by 0.
The final conditional we will look at today is known as the substitution property, and it is incredibly useful in proofs.
If two values are equal, then they may substitute for each other.
For example, suppose we know x=y, and that x+2=4. Then we also know by the substitution property that y+2=4, since x and y can substitute for each other.
It is very important that you remember the names of these conditionals. The four "properties of equality" should be easy to remember, since they are named for the arithmetic operation that they have in their property. The substitution property also should be easy to remember, since you substitute values. That just leaves the transitive, symmetric, and reflexive property. You can remember the Transitive property if you remember that it uses Three values. If you remember lines of symmetry, then you can say that the symmetric property says that equations have lines of symmetry right down their equals sign. As for reflexive? Well, you'll just have to remember it.
Also note that you can use other properties from algebra, for example
a(b+c)=ab+ac by the distributive property.

Proof

Given: ${\overline {AC}}\cong {\overline {BD}}$ Prove: ${\overline {AB}}\cong {\overline {CD}}$ Statements Reasons
1. ${\overline {AC}}\cong {\overline {BD}}$ 1. Given
2. AC=BD 2. Definition of congruent segments
3. AB+BC=AC; BC+CD=BD 3. Segment addition postulate
4. AB+BC=BC+CD 4. Substitution property
5. BC=BC 5. Reflexive property
6. AB=CD 6. Subtraction property of equality
7. ${\overline {AB}}\cong {\overline {CD}}$ 7. Definition of congruent segments
8. Q.E.D.