Warm-up: Fill in the blank with always, sometimes, or never.
1) Vertical angles are ______ congruent.
2) Congruent angles are _______ right angles.
3) Complementary angles are ______ congruent.
4) Adjacent angles in a transversal are _______ supplementary.
5) Linear pairs are ________ supplementary.
6) Angles that are supplementary _________ add up to 180º.
7) Supplementary angles are ________ a linear pair.
1. always 2. sometimes 3. sometimes 4. always 5. always 6. always 7. sometimes
Two forms of a sentence in a proof:
Since this condition is met, I can conclude this fact.
I can conclude this fact because this condition is met.
Rewrite each statement in its alternative form:
a. I am going to mow the grass because the grass is too high.
b. Since I’m being paid $10, I am going to mow the grass.
Often times, the idea of a =b, b=c, so a=c so comes so easily to us, that we forget to point out how our minds actually reach these conclusions.
For example, many of you go right from angles being a linear pair to the conclusion that they add up to 180, and while you are right, the actual mathematical reasoning goes like this:
a.) ‘Linear pairs are supplementary angles by the Linear Pair Theorem, supplementary angles add up to 180º by the definition of Supplementary angles, so linear pairs add up to 180º.’
This is because the Linear Pair Theorem states: Linear pairs are supplementary.
If the Linear Pair Theorem stated: “Linear pairs add up to 180 degrees,” our knowledge of facts about linear pairs would flow like this:
b.) ‘Linear pairs add up to 180º by the Linear Pair Theorem, angles that add up to 180º are supplementary by the definition of supplementary angles, so linear pairs are supplementary.’
But since that first statement in part b is not what the Linear Pair Theorem actually gives us, we go with the line of thinking in part a.
Now, use the given facts in scenarios 1 and 2 below to draw a valid conclusion, and show your thinking in the form of a “Since I know, then I can say” statement and as an “I can say, because I know…” statement.
1. The trash is collected on Friday mornings, trash is put out the night before it is collected, Mr. Z. takes out the trash.
2. Mr. Z. takes out the trash on Thursday nights, the trash is put out the night before garbage is collected.
Create your own situation with given information that allows you to draw a certain conclusion. Think about these categories: School, grades, work, shopping, food, sports, chores, rules, laws, interactions with people, likes/dislikes, videogames, games, science, history, English.
The whole point of these exercises is to acknowledge the thinking that you actually do before you reach a conclusion. Proving statements is just the process of showing every little step, leaving no thought out for someone else to have to fill in.