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Chapter 3: Parallel and Perpendicular Lines

. . Chapter 3: Parallel and Perpendicular Lines. Pages 113 - 177. Ch.3 (part 1). 3-1 Properties of Parallel Lines. p. 115. transversal - a line that intersects two or more lines in a plane at different points. t. Line t is a transversal. p. m.

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Chapter 3: Parallel and Perpendicular Lines

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  1.  Chapter 3: Parallel and Perpendicular Lines Pages 113 - 177

  2. Ch.3 (part 1) 3-1 Properties of Parallel Lines p. 115 • transversal - a line that intersects two or more lines in a plane at different points. . t . Line t is a transversal. p m * Pairs of the eight angles have special names as suggested by their positions.

  3. Transversals and Angles p Ex.] Transversal p intersects lines s and t. 1 2 4 3 s exterior angles: 1, 2, 7, 8 5 6 t 8 7 interior angles: 3, 4, 5, 6 corresponding angles: 1 and 5, 2 and 6 4 and 8, 3 and 7 consecutive (same-side) interior angles: 3 and 6, 4 and 5 alternate interior angles: 4 and 6, 3 and 5 alternate exterior angles: 1 and 7, 2 and 8

  4. 2 1  3 4 5 6  7 8 ~ ~ ~ ~ Ex.] 1 = 5, 2 = 6, 3 = 7, 4 = 8 Postulate 3-1 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

  5. Thm. 3-1Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then each pair of alternate interior angles is congruent. Ex.] 4 = 5, 3 = 6 ~ ~ Thm. 3-2Consecutive (same-side) Interior Angles Theorem If a transversal intersects two parallel lines, then each pair of same-side interior angles is supplementary. Ex.] m4 + m6 = 180 m3 + m5 = 180 2 1  3 4 5 6  8 7

  6. p. 117 • two-column proof - the statements and reasons are aligned in columns Given: ~~~~~ (what you know) picture (sometimes) Prove: ~~~~~ (what you must show) Statements Reasons 1. ~~~~~~ 1. ~~~~~~ 2. ~~~~~~ 2. ~~~~~~ 3. ~~~~~~ 3. ~~~~~~

  7. ~ ~ Examples: If 3 = 6 , or if 4 = 5 , then m  n.  2 1 3 4 5 6 8  Examples: If 3 + 5 = 180, or if 4 + 6 = 180, then m  n. 3-2 Proving Lines Parallel p. 122 Thm. 3-3 Converse of the Alt. Int. Angles Thm. If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Abbreviation: If alt. int. s are =, then lines are  . ~  p m n 7 Thm. 3-4 Converse of the Consec. Int. Angles Thm. If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel.

  8. ~ ~ ~ Examples: If 1 = 5 , 2 = 6 , 3 = 7, or 4 = 8 , then m  n. ~  2 1 3 4 5 6 8 Postulate 3-2 Converse of the Corr. Angles Post. If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel. Abbreviation:If corr. s are =, then lines are  . ~  p m n 7 • flow proof - arrows show the logical connections between statements

  9. Thm. 3-5 If two lines are parallel to the same line, then they are parallel to each other. b   Examples: If b  d and c  d, then b c.  c    d Thm. 3-6 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Abbreviation: If 2 lines are  to the same line, then lines are . b Examples: If b  c and b  d, then c d.  c d 

  10. 3-3 Parallel Lines and the Triangle Angle-Sum Theorem p. 131 Thm. 3-7 Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180. C Examples:mA + mB + mC = 180 A B - You can classify triangles by angles…. • equiangular - angles are all congruent • acute - triangle with all acute angles • right - triangle with one right angle • obtuse - triangle with one obtuse angle

  11. …and you can classify triangles by sides. • equilateral - triangle with all sides congruent • isosceles - triangle with at least two sides congruent • scalene - triangle with NO sides congruent • exterior angle of a polygon - an angle formed by a side and an extension of an adjacent side Q Examples:QRT is an exterior angle of SQR. T 40° 140° S R

  12. remote interior angles - the two nonadjacent interior angles corresponding to each exterior angle of a triangle 1 Examples:1 and 2 are remote interior angles of 3 2 3 Thm. 3-8 Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. Examples:m3 = m1 + m2 (use picture above)

  13. 3-4 The Polygon Angle-Sum Theorems p. 143 • polygon - closed plane figure with at least three sides that are segments • - The sides intersect only at their endpoints and no two adjacent sides are collinear. • - The vertices are the endpoints of the sides. • - A diagonal is a segment that connects two nonconsecutive vertices. • - convex - no diagonal contains a point outside the polygon • - concave- a diagonal contains points outside that polygon (Examples on board)

  14. - You can classify a polygon by the number of sides it has.

  15. Thm. 3-9 Polygon Angle-Sum Theorem The sum of the measures of the interior angles of an n-gon is (n - 2)180. Example:Find the sum of the measures of the angles of a 13-gon. Sum = (n - 2)180 = (13 - 2)180 = 1980 Thm. 3-10 Polygon Exterior Angle-Sum Theorem The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. 1 4 Examples:m1 + m2 + m3 + m4 = 360 (for a square) 2 3

  16. equilateral polygon - a polygon with all sides congruent • equiangular polygon - a polygon with all angles congruent • regular polygon - a polygon that is both equilateral and equiangular

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