Parallelogram Proofs

Parallelogram Proofs

By Rachel McKaughan

​Draw the Givens in BLACK

​Draw Givens in BLACK
Draw the STATEMENT we need to prove in RED

Multiple Choice

What STATEMENT are we trying to prove?

• AB // DC

• AD // BC

​Circle the p (REASON) part and underline the q (STATEMENT) part

​If you have PROOF of your REASON (p), then you can write the STATEMENT (q) you need.

​Logic = If p, then q.

​Draw Givens in BLACK
Draw the STATEMENT we need to prove in RED

​Using the Alternate Interior Angle Converse Theorem as our REASON, we can write a STATEMENT that two lines are parallel (which is what we need to do) if they are cut by a transversal that has congruent alternate interior angles.

​In GREEN, draw over the two lines that we need to prove are parallel AND the transversal that cuts across them making congruent alternate interior angles.

Do we have a REASON to justify the STATEMENT that AD // BC?

​Givens: Black
Statement we need to prove: Red

Poll

Have we proven that AD // BC with the Alternate Interior Angles Converse?

• yes

• no

​Time to fill in STATEMENT #3 that we were able to PROVE using the REASON that Alternate Interior Angles Converse gave us (If p, then q).

proofs = logic aka REASONing

​If you have PROOF of your REASON (p), then you can write the STATEMENT (q) that you need in order to REASONably prove the next STATEMENT.

​When we write our REASONS, we write the name of the Theorem or Definition or Postulate or Property.

​Definitions are BIconditional

If p, then q AND if q, then p.
If a, then b AND if b, then a.

​Theorems only go ONE WAY:
If p, then q. If a, then b.





That's why there can be theorem CONVERSES: If q, then p. If b, then a. (but not for all) (vert)

NOTE: reflexive property and similar triangles.

Draw the triangles or write a similarity statement to get the segments written correctly.

• ​Givens: Black

• Need to prove: Red

• Let's work backwards and see what we can prove.
  

​If REASON, then STATEMENT Since we have the reasons already, what statements do they prove?
If they don't prove it yet, what are we missing?

• Now let's look at it from the beginning.

• Given a couple of facts, we have proven this shape is a parallelogram based on logic.

• One step at a time, each step built on top of a solid foundation of facts (statements) proven by logical (reason)ing. "Since these previous statements are true, I can say the next statement is true, based on this reason."