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Chapter 6: Parallel Lines

Chapter 6: Parallel Lines. By: David and AJ. Lesson 1: Line Symmetry. Theorem 16 In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment. Definitions:

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Chapter 6: Parallel Lines

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  1. Chapter 6: Parallel Lines By: David and AJ

  2. Lesson 1: Line Symmetry Theorem 16 In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment Definitions: Two points are symmetric with respect to a line iff the line is the perpendicular bisector of the line segment connecting the two points

  3. Most construction proofs involve Congruent triangles Construction To construct a line perpendicular to a given line through a given point Point on a line Point not on a line Perpendicular Bisector

  4. Lesson 2: Proving Lines Parallel Two lines are Parallel iff they lie in the same plane and do not intersect

  5. Transversal A Transversal is a line the intersects two or more lines in different points

  6. Corresponding Angles When a transversal intersects two lines in the same plane, it forms pairs of angles that are given special names Interior Angles on the same side of the transversal Alternate Interior Angles

  7. Theorems and Corollaries Theorem 17 Equal corresponding angles mean that the lines are parallel Corollary 1: Equal alternate interior angles mean that the lines are parallel. Corollary 2: Supplementary interior angles on the same side of a transversal mean that the lines are parallel. Corollary 3: In a plane, two lines perpendicular to a third are parallel.

  8. Lesson 3:The Parallel Postulate

  9. Construction To construct a line parrallelto a given line through a given point

  10. Boring Stuff Postulate 7 Through a given point not on a line, there is exactly one line parallel to the given line

  11. Theorem 18 In a plane, two lines parallel to a third line are parallel to each other

  12. Lesson 4: parallel lines and angles Theorem 19: Parallel lines form equal corresponding angles AB llCD a = e b = f c = g d = h

  13. Corollaries to theorem 19: #1: parallel lines form equal alternate interior angles M ll L x = x y = y

  14. #2: Parallel lines form supplementary interior angles on the same side of a transversal

  15. #3: In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other k ll l t perp k t perp l

  16. Lesson 5: The angles of a triangle Theorem 20: The Angle Sum Theorem The sum of the angles of a triangle is 180 degrees DUH!!

  17. Corollaries to theorem 20 #1: If two angles of one triangle are equal to two angles of another triangle, the third angles are equal

  18. Corollary #2: The acute angles of a right triangle are complementary

  19. Corollary #3: Each angle of an equilateral triangle is 60 degrees Why? Because: 180/3=60

  20. Theorem 21: The Exterior Angle Theorem An exterior angle of a triangle is equal to the sum of the remote interior angles

  21. Lesson 6: AAS and HL congruence Lesson 6: AAS and HL congruence

  22. Theorem 22: The AAS Theorem If two angles and the side opposite one of them in one triangle are equal to the corresponding parts of another triangle, the triangles are congruent

  23. Theorem 23: The HL Theorem If the hypotenuse and a leg of one right triangle are equal oh corresponding parts of another right triangle, the triangles are congruent

  24. PRACTICE PROOFS G 1 H U 2 3 S Given: 1 and 2 are Complementary 2 and 3 are complementary Prove: GU ll HS

  25. PRACTICE PROOFS G H U Given: GU ll HS US ll GH Prove: triangle GUS is congruent to triangle SHG S • Be sure to name triangles in same order of points

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