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Parallel and Perpendicular Lines

Parallel and Perpendicular Lines. Geometry – Chapter 3. Chapter 3 Standards . 2.0 Students write geometric proofs 4.0 Students prove basic theorems involving congruence 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal

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Parallel and Perpendicular Lines

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  1. Parallel and Perpendicular Lines Geometry – Chapter 3

  2. Chapter 3 Standards • 2.0 Students write geometric proofs • 4.0 Students prove basic theorems involving congruence • 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal • 12.0 Students find and use measures of interior and exterior angles of triangles and polygons to classify figures and solve problems.

  3. 3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines. • a transversal is a line that intersects two coplanar lines at two distinct points. • the diagram shows the eight angles formed by a transversal t and two lines, l and m.

  4. 3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines. • There are special names for certain angles in a 2 line and transversal relationship • alternate interior angles • same-side interior angles • corresponding angles • alternate exterior angles • same-side exterior angles • Draw a transversal using a ruler through your not-parallel lines. Discuss which angles you think are which and why.

  5. 3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines. • alternate interior angles • same-side interior angles • corresponding angles • alternate exterior angles • same-side exterior angles

  6. 3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines. • Draw a transversal using a ruler through your parallel lines. Use a protractor to measure all of the angles. Discuss and draw conclusions about angle relationships when the two lines are parallel. • alternate interior angles • same-side interior angles • corresponding angles • alternate exterior angles • same-side exterior angles

  7. 3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines.

  8. 3.1 Properties of Parallel LinesEQ: Identify different types of angles formed by parallel lines. • Homework: page 132 (1-16) all

  9. 3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1 • What is the Corresponding Angles Postulate? • What is the converse to this?

  10. 3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1 • Everything goes back to either the Corresponding Angles Theorem or the Converse of the Corresponding Angles Theorem. • When you begin a proof involving parallel lines, you should ask yourself “How do I show that corresponding angles are congruent?”

  11. 3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1 • What is the converse to the Alternate Interior Angles Theorem? • If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. • Proof:

  12. 3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1 • What is the converse to the Same-side Interior Angles Theorem? • If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. • Proof:

  13. 3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1 • State the Converse to the Alternate Exterior Angle Theorem • If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. • Proof:

  14. 3.2Proving Lines ParallelEQ: Prove the converses to the theorems of section 3.1 • State the Converse to the Same-side Exterior Angle Theorem • If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. • Proof:

  15. 3-3 Parallel and Perpendicular LinesEQ: Use previously proven theorems to prove theorems about parallel and perpendicular lines.

  16. Homework: • page 137 (1-21) all • page 143 (1-3)

  17. Warm Up • In this drawing, line k is parallel to line j • Which angle is alternate interior with ∠4? • Which angle is corresponding to ∠8? • m∠3 = 37. What is m∠6? • m∠1 = x+12 and m∠5 = 3x – 36. What is x? • Given that k∥j, write a proof to show that ∠2 and∠5 are supplementary.

  18. 3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

  19. 3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

  20. 3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

  21. 3-4 The Triangle Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a triangle.

  22. 3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

  23. 3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

  24. 3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

  25. 3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon.

  26. 3-4 The Polygon Angle Sum TheoremEQ: Determine the measures of interior and exterior angles of a polygon. • homework: • page 150 (1-6, 10-20) all • page 161 (1-21) all

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