2 5 Conjectures that Lead to Theorems Obj

2. 5 Conjectures that Lead to Theorems Obj: Understand use vertical angle theorem

Why do we need Proofs? ? ? 1 region 1 4 regions 8 regions 2 2 regions 16 regions

How many regions will be in a circle with 6 pts. ? ? ? Try it! Only 31

So we need to prove conjectures because sometimes what seems to make sense is not always true!!!

<1 and <4, <2 and <3 are vertical angles. How would you describe the relationship of these angles? 1 2 3 4 Vertical angles: the opposite angles formed by 2 intersecting lines. <1 and <4 are vertical angles.

Discovering a Theorem What is the measure of the angle?

Theorem l Vertical Angle Theorem: if 2 angles form a pair of vertical angles, then they are congruent

Given: <1 and <2 are vertical angles Prove: <1 <2 Statement 1 3 4 2 Reason

Practice

The m<TSU is 80, and the m<USP is 60 What is the measure of the other 4 angles?

Congruent Supplements Theorem l If 2 angles are supplements of congruent angles, then the 2 angles are congruent. <A is the supplement of <B <C is the supplement of <B Therefore, _______ ¡ We will prove this in our homework p. 123 (25 -27)

Chapter 2 Topics l Intro to Proof ¡ Inductive and deductive reasoning ¡ What is a proof? l Intro to Logic ¡ Conditional ¡ Hypothesis and conclusion ¡ Euler diagram ¡ Converse ¡ Counterexample ¡ Laws of logic

l Definitions ¡ Biconditionals ¡ Determining if a statement is a definition ¡ Adjacent angles l Properties of equality and congruence ¡ Two-column and paragraph proofs ¡ Overlapping angle thm. ¡ Overlapping angles thm l Conjectures leading to thms ¡ Linear pairs ¡ Vertical angles ¡ Congruent supplements thm.

Should have 5 sheets of Notes l Any material on notes is fair game for the test l Titles of note sheets ¡ 2. 1 Intro to Proof ¡ 2. 2 Intro to Logic ¡ 2. 3 Definitions ¡ 2. 4 Building a system of Geo Knowledge ¡ 2. 5 Conjectures that lead to thms