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Geometry Day 20

Geometry Day 20. Proving Lines are Parallel. Today’s Agenda. Proving lines parallel through: Corresponding angles Alternate Interior angles Same Side Interior angles Alternate Exterior angles Using inductive and deductive reasoning to develop Geometric concepts. Review.

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Geometry Day 20

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  1. Geometry Day 20 Proving Lines are Parallel

  2. Today’s Agenda • Proving lines parallel through: • Corresponding angles • Alternate Interior angles • Same Side Interior angles • Alternate Exterior angles • Using inductive and deductive reasoning to develop Geometric concepts.

  3. Review Parallel lines and transversals form special angles: • Corresponding angles are congruent • Alternate interior angles are congruent • Same side interior angles are supplementary

  4. Be Able to Name Special Pairs of Angles: • Alternate Interior Angles • Corresponding Angles • Same-Side Interior Angles 2 1 4 3 6 5 7 8 • Be Able to State the Relationship Between Any Two Angles: • Congruent Angles • Supplementary Angles

  5. Converse of Corresponding Angles Postulate (p. 205) • Remember, just because a conditional statement is true, we cannot assume that its converse will be true. We have to examine it separately. • The converse of the corresponding angles postulate states that if two lines are cut by a transversal, and the corresponding angles are congruent, then the lines are parallel. • Do we accept this?

  6. Another Parallel Postulate (p. 206) • If given a line and a point not on the line, there is exactly one line that passes through the point and is parallel to the line. • This one is controversial.

  7. More converses (p. 206) • The converses of the other angle pairs are theorems: • If alternate interior angles are congruent, then the lines are parallel. • If same side interior angles are supplementary, then the lines are parallel. • If alternate exterior angles are congruent, then the lines are parallel. • If two coplanar lines are both perpendicular to the same line, then they are parallel.

  8. Remember!!! • Do not get the original theorems mixed up with the converses! • The original theorems start with parallel lines as a given, and we can conclude that the angle pairs are congruent or supplementary. • If the lines aren’t given as parallel, you can’t assume that the angle relationships will be true! • But if the angles have the proper relationship, then we can conclude that the lines are parallel.

  9. Constructions • Review – How do you construct: • a congruent line segment? • congruent angles? • an angle bisector? • a segment bisector?

  10. Constructions • Parallel line through a given point • This construction is based on the converse of the corresponding angles postulate • Watch video

  11. Homework 12 • Workbook pp. 39-40

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