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Understanding Vertical Angle Theorem and Proofs in Geometry

Learn and apply the Vertical Angle Theorem and why proofs are necessary in geometry. Explore the relationship of vertical angles and discover the congruence concept. Introduction to proof, logic, and key geometry topics. Sheets of notes included.

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Understanding Vertical Angle Theorem and Proofs in Geometry

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  1. 2.5 Conjectures that Lead to Theorems Obj: Understand and use vertical angle theorem

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

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

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

  5. <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.

  6. Discovering a Theorem What is the measure of the angle?

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

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

  9. Practice

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

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

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

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

  14. Should have 5 sheets of Notes • Any material on notes is fair game for the test • 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

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