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Chapter 6: Parallel Lines

Chapter 6: Parallel Lines. Matt Palmer. Lesson 1: Line Symmetry. Two point are symmetric with respect to a line if and only if the line is the perpendicular bisector of the line segment connecting the two points. Must be right angle. Theorems.

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Chapter 6: Parallel Lines

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  1. Chapter 6: Parallel Lines Matt Palmer

  2. Lesson 1: Line Symmetry • Two point are symmetric with respect to a line if and only if the line is the perpendicular bisector of the line segment connecting the two points Must be right angle

  3. Theorems • Theorem 16: In a plane, 2 points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment.

  4. Lesson 2: Proving Lines Parallel • Parallel- Two lines that lie in the same plane and do not intersect • Transversal- A line that “intersects” two or more lines in different points

  5. More Key Terms • Corresponding Angles- Form an F • Alternate Interior Angles- Form a Z • Same Side Interior Angles- Form a U

  6. Theorems and Corollaries • Theorem 17- Equal corresponding angles will form parallel lines. • Corollary 1- Equal alternate interior angles will form parallel lines. • Corollary 2- Supplementary same side interior angles will form parallel lines.

  7. Lesson 3: The Parallel Postulate • The Parallel Postulate- Through a point not on a line, there is exactly one line parallel to the given line.

  8. Theorems • Theorem 18- In a plane, 2 lines parallel to a third line are parallel to each other. Because A is parallel to B, and B is parallel to C, then A is parallel to C

  9. Lesson 4: Parallel Lines and Angles • Theorem 18- Parallel Lines will form equal corresponding angles. • Corollary 1- Parallel lines form equal alternate interior angles. • Corollary 2- Parallel lines for supplementary same side interior angles.

  10. Quick Quiz on Parallel Lines Angles are not 90 degrees  Equal Angles? Equal Angles? Supplementary Angles? Supplementary Angles?

  11. Lesson 5: Angles of a Triangle • Angles Sum Theorem- The sum of the angles of a triangle is 180 degrees. • Theorem 21- An exterior angle of a triangle is equal to the sum of the remote interior angles Angle W = X +Y

  12. Angle Sum Theorem

  13. Corollaries • Corollary 1- If 2 angles of one triangle are equal to 2 angles of another triangle, then the third angle is equal. • Corollary 2- The acute angles of a right triangle are complementary. • Corollary 3- Angles of equilateral triangles are equal to 60 degrees.

  14. Lesson 6: AAS and HL Congruence • AAS Theorem- If 2 angles and the side opposite one of them in one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.

  15. HL Theorem • HL Theorem- If the hypotenuse and a leg of one right triangle are equal to the corresponding parts of another right triangle are equal, then the triangles are congruent.

  16. Proofs • The Proofs from this chapter were all about either proving lines parallel or segments to be congruent.

  17. Example 1 Given: 1 and 2 are complementary; 2 and 3 are complementary. Prove: GU || HS G 1 2 U S 3 H

  18. Example 2 Given: 1 and 2 are vertical angles; B is the midpoint of LW; EL WO. Prove: EL = OW L 1 B O E 2 W

  19. Chapter 6 LabProving Geometric Constructions • This lab was to prove the reasoning behind different postulate or theorms.

  20. Example From The Lab • This picture proves that by construction you can create an angle bisector from just an angle and a protractor. WOW!

  21. Insight! • Make sure you know which angle pair needs to be equal to create parallel lines and which one needs to be supplementary. • Memorize the angle pairs by the letter names on slide 5 • DON’T GET FRUSTRATED! Some of these problems (pictures) take a little while to see the main point. • Thanks.

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