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Math 20-2 Geometry: Lesson #2 Proving Properties of Parallel Lines & Transversals Objective: By the end of this lesson, you should be able to: - Prove, using deductive reasoning, properties of angles formed by transversals and parallel lines. - Identify and correct errors in a given proof of a property that involves angles. Last class, we made conjectures about the relationships between angles formed by parallel lines and a transversal, using inductive reasoning. Remember: Inductive reasoning does not prove that something is true. This lesson, we will be using deductive reasoning to prove the angle properties. Vocabulary: Supplementary angles – The type of proof that we will use most often in this unit is called the Two-Column Proof. A two-column proof looks like this: Statement Reason - - True mathematical (geometric) facts Reason that you know each statement is true The “Reasons” in these proofs can be the reasons we gave in the Warm-Up: - opposite angles (equal) - alternate angles (interior or exterior) (equal) - corresponding angles (equal) - interior angles (supplementary) Another common “Reason” is the Transitive Property: To prove a statement, you may use other properties of the angles formed by parallel lines and a transversal, but you may not use the property that you are trying to prove. This would be using circular reasoning, which is not a valid method of proof. Math 20-2 Geometry: Lesson #2 e.g. 1) Prove that opposite angles are always equal. Statement Reason e.g. 2) Prove that when a transversal intersects parallel lines, the alternate interior angles are equal. Statement Reason The converse of the statements about angles formed by parallel lines and a transversal are also true. If: alternate angles (interior or exterior) are equal, OR corresponding angles are equal, OR the interior angles on the same side of the transversal are supplementary then the lines are parallel. This means that in a proof, you can use any one of these statements as the reason for stating that two lines are parallel. Math 20-2 Geometry: Lesson #2 e.g. 3) In the diagram below, CBD BDE . Prove that ABE BED . C Statement Reason D B A E e.g. 4) Identify and correct the error in the proof that Z VWX . Given: VWX VXW WX YZ V W Y Assignment: X Z p. 78-82 #1-3, 8, 12, 16, 19 Statement Reason VWX VXW Given VXW Z Alternate interior angles Z VWX Transitive property