Logic and Parallel Lines Ms. Wightman’s Geometry Unit, Lesson 1

Logic  Logic-[loj-ik] Noun. Reason or sound judgment. (  Why do we need logic?  Ex: “just because” always seems unfair  We use logic to create sound (or correct) proofs that support our understanding of mathematics.

Parts of Proofs  Given statement- This statement is something we know or are assuming is true.  Statement- This is a piece of information we are going to use to deduce new information.  Deduce- to reach a conclusion  Conclusion- The piece of information we know to be true based off of our given statement, statement, and logic.

Logic: Examples  Given Statement: If Denise has 20 or more cats, then she is considered a cat lady.  Statement: Denise has 30 cats.  Conclusion: Denise is a cat lady.  Statement: Denise has 19 cats.  Conclusion: Denise is not a cat lady.  Statement: Denise is not a cat lady.  Conclusion: Denise has 19 or less cats.

A Quick Refresher on Proofs  We’re going to take the statement “If Denise has 20 or more cats, then she is considered a cat lady,” as a theorem.  Prove that if Denise has 30 cats, she’s a cat lady. Statement Reason 1.Denise has 30 cats. 2.Denise has more than 20 cats. 3.Denise is a cat lady. 1.Given. 2.Since 30 > By our “theorem”.

Parallel Lines

Definition  What do you think parallel means?  You’ve seen parallel lines before…can you think of where? Football Field Parallel Parking After seeing these real world examples…can you think of a mathematical definition of parallel?

Definition  Parallel- two lines l and m are parallel provided that if extended indefinitely, they never intersect. We write l || m.  Transversal- A transversal t is a line that intersects two (nonequal) parallel lines Transversal Parallel lines

Corresponding Angles  What’s another word for correspond?  Match up with, coincide, etc. List of corresponding angles: <AEG and <CFE <GEB and <EFD <AEF and <CFH <BEF and <DFH

(Watch video to 4:59)

What can you guess is true about corresponding angles? Discuss with your partner. Write down your answer in your guided notes.

Theorem Claim: When a transversal crosses parallel lines, corresponding angles are congruent. With your partner, try to prove this theorem. (Hint: Triangles are a good start, remember SAS, SSS, ASA!)

Alternate Interior Angles  Interior Angles- angles that are between the two parallel lines and lie on the transversal  Alternate Interior Angles- interior angles on one parallel line that correspond to interior angles on the other parallel line <AEF and <EFD are alternate interior angles; <BEF and <EFC are alternate interior angles; All mentioned angles are interior angles.

What can you guess is true about alternate interior angles? Discuss with your partner your ideas. Write down your answer in your guided notes.

Theorem Claim: When a transversal crosses parallel lines, alternate interior angles are congruent. With your partner, try to prove this theorem. (Hint: Triangles are a good start!)

Homework In this class, you will be creating your own “Proof Book” that acts as a record for every major theorem we prove in class. I will let you know each day which theorems need to be in the Proof Book. You can design or decorate the Book in any way you like, as long as the assigned proofs are in the Book. For tonight’s homework, copy down the two proofs we did in class today.