There is a lot of prior information that goes into understanding proofs!
Let’s start with some real life examples, with the idea of conditions.
What are the conditions under which you would mow the lawn?
Ex: “I would mow the lawn under the conditions that I get payed at least $10.”
Here, the condition is that you be payed $10, and if that condition is met, then you will mow the lawn. If there is no $10, forget about it.
“I would mow the lawn under the conditions that the grass is too high.” Here, the condition is that “the grass must be too high”. If the grass is not too high, you won’t mow the lawn.
Also notice that these kinds of statements, called conditional statements, can be rewritten in an “If… then…” form. “If I get payed $10, then I will mow the lawn.” Notice that the part of the statement following the word “if” is the condition of the statement, getting paid $10.
Now we move on to the idea of theorems:
Mr. Z’s Thursday Trash Theorem- On Thursday nights, Mr. Z. takes the trash to the curb.
What are the conditions here? This theorem is only relevant when it is a Thursday night.
What is the conclusion we can draw if it is in fact a Thursday night? Mr. Z. must have taken the trash to the curb.
Once again, we can rewrite this as an “If… then…” statement: If it is Thursday, then Mr. Z. will take the trash to the curb.
Consider this next theorem: Mr. Z’s Wednesday Dishes Theorem
“On Wednesdays when the dishes are dirty, Mr. Z. will wash them”
Here, we have two conditions: 1.) It is Wednesday and 2.) The dishes are dirty. For Mr. Z. to wash dishes, both of these conditions must be met. So, if it is Wednesday and the dishes are dirty, then Mr. Z. will wash the dishes.
Now, let’s remind ourselves of the actual mathematical theorems postulates, properties, and definitions that we will be dealing with and using.
Corresponding Angles Postulate: If a transversal crosses two parallel lines, the corresponding angles created are congruent.
Vertical Angles Theorem: Vertical angles are congruent.
Linear Pair Theorem: Linear pairs are supplementary.
Supplementary- The measures of angles that are supplementary have a sum of 180 degrees.
Congruent Angles- Angles that are congruent have equal measures.
Substitution Property- If a = b, then a can be used in place of b in any equation or inequality, or b can be used in place of a in any equation or inequality.
Now, here’s an example of using these properties to prove statements.