Group Class Activity (Friday 10/14)

Definition: A quadrilateral is a parallelogram if both pairs of opposite sides are parallel.

(1) On a page, write the names of the members of your group.

(2) Prove that the following statements are equivalent. Write down the proofs neatly and in order. (See the notes on the Tools page on the back).

(3) Write the proofs in a correct logical sequence. Take turns writing the proofs even though the reasoning should be a collaborative effort; each person should initial each proof s/he wrote.

(4) Draw a diagram with numbers 1, 2, 3, 4, 5 and then draw arrows from one number to another showing any "if, then" statement you proved.

1. A quadrilateral is a parallelogram. (This means it satisfies the definition of parallelogram.)

2. A quadrilateral has both pairs of opposite sides congruent.

3. A quadrilateral has both pairs of opposite angles congruent

4. A quadrilateral has one pair of sides parallel and congruent.

5. The diagonals of the quadrilateral bisect each other. (This means that the point of intersection of the diagonals is the midpoint of each diagonal.)

Tools and hints

Tools: What you know or can assume in your proofs.

You can assume BB Principles 1-12 and also Theorems 13-16. Theorems 14 and 15 will be especially important.

Important: Once you prove something, you can use it (carefully).

• Once you should two statements are equivalent, for proofs that come later you can use the two interchangeably.
• More subtly, suppose you prove "if 2 then 3" and you want to prove "if 2 then 4". You can start by assuming 2, then you can say that 3 is true because 2 is assumed true, so you can then use both 2 and 3 to prove 4.

Hints: Proving equivalence

Proving 1, 2, 3, 4, 5, equivalent means proving "#1 if and only if #2", and "#2 if and only if #4", etc. for all possible combinations.

You can save a lot of work by building some chains of implication. For example, if there were four statements, p, q, r, s, then if you proved "if p, then q" and "if q, then s" and "if s, then p", you would have "p if and only if q if and only if s". So p, q, s are equivalent. Then if you could prove "if p, then r" and "if r, then a" you would (using the already proved "if q, then p") another chain making p, q, r equivalent, so that p, q, r, s are equivalent.