**Geometry Chapter 2**

Objectives

Conditional Statements

Conjectures

Math History:

Proofs

Geometric proofs date back a very long way. Two Ancient Greek mathematicians named Exodus and Theaetetus drew up theorems but did not prove them. The Greek philosopher Thales proved geometric theorems, however, he didn't use two-column proofs as we do in math class. Finally, a man named Euclid of Alexandria used undefined terms,

which were self-evidently true, and deductive logic to prove theorems true. He transformed proofs with that, and Islamic mathematicians advanced them a little further before the proof theory we use today took shape. Axioms, which are statements that suggest that undefined terms are true, are no longer assumed true, allowing for more mathematical theories to be built.

**By: Emily, Micaela, and Reilly**

Chapter 2 of Geometry is mainly about reasoning and geometric proofs. This chapter covers topics like conjectures, inductive/deductive reasoning, logic, conditional statements, and algebraic or two-column proofs. Today, we're presenting Chapter 2 in just a few minutes, but given our presentation, we can prove that you'll remember all of the content by the end!

Chapter 2: Reasoning and Proofs

Mathletics

1. If an animal is spotted, then it is a Dalmatian.

Choose the hypothesis of the following statements

2. If today is Thanksgiving, then today is Thursday.

Write each statement in if-then form

3. Cheese contains calcium.

4. The measure of an acute angle is between 0 and 90.

Write the converse, inverse, and contrapositive of each conditional statement.

6. If a triangle is a right triangle, then it has one right angle.

5. If two angles are supplementary, then they each measure 90.

Draw the object that would come next in the following pattern.

7.

Students will review:

Conjectures

Conditional Statements

Law of Detachment and Syllogism

Proofs

Possible answers:

Get a free phone with a two-year service contract.

Equiangular triangles are equilateral.

Answers:

If you get a two-year service contract, then you get a free phone.

If a triangle is equiangular, then it is equilateral.

If p, then q

P is hypothesis

Q is conclusion

Converse (if q, then p)

Inverse (if not p, then not q)

Contrapositive (if not q, then not p)

Biconditional (p if and only if q)

How they Work

C

D

A

B

K

Put the following statements into if-then form.

What is the converse of the statement "Vertical angles are congruent."

Conditional Statements cont.

Conjectures cont.

Educated guess

Can be true or false

False need a counterexample

Conditional Statements cont.

Answer: If angles are congruent, then they are vertical

Different conditionals have different truth values

Show in truth table

Converse and inverse correlate

Conditional and contrapositive correlate

Detachment & Syllogism

1. A midpoint divides a segment into 2 congruent parts

2. C is the midpoint of RS

3. RC is congruent to CS

Which Law?

If p then q is true, and p is true, then q is true

Two variables

Detachment

Laws cont.

True: Angle AKC is congruent to Angle BKD.

False: Line AC is perpendicular to line BD.

Counterexample: m<AKC = 87 degrees

Syllogism

1. If we have over 4 in. of snow, the buses cannot drive.

2. If the buses cannot drive, school will be canceled.

3. ???

Answer: If we have over 4 inches of snow, school will be canceled.

IF p then q is true AND if q then r is true THEN p then r is true.

Must follow the certain order

Wrong Order= NO possible conclusion

Proofs... BREIFLY!

Postulates

and

Theoroms

are used in proofs

Postulates

are accepted

Theoroms

are proven

Properties

are also used

Ex. Addition Prop, Transitive Prop.