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USING PARALLEL AND PERDENDICULAR LINES

USING PARALLEL AND PERDENDICULAR LINES. OBJECTIVES. - Solving problems by using a diagram -Use properties of parallel lines -Use slope to identify parallel and perpendicular lines -Prove lines parallel -Apply distance relationships among points, lines, and planes. Proofs.

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USING PARALLEL AND PERDENDICULAR LINES

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  1. USING PARALLEL AND PERDENDICULAR LINES OBJECTIVES -Solving problems by using a diagram -Use properties of parallel lines -Use slope to identify parallel and perpendicular lines -Prove lines parallel -Apply distance relationships among points, lines, and planes

  2. Proofs • Proofs are built on ‘if—then’ statements • ALWAYS make a diagram of the data & mark the ‘givens’ • The givens are ‘ifs’ • Start with ‘if’ and find a postulate, theorem or definition that will take you to the next step. • The last step is what was to be proven. • Transitivity & substitution are often used for justification • An example follows at the end of the power point.

  3. Types of Lines • Parallel lines—same plane, never cross • Skew lines—different planes, never cross • Traversal lines –same plane, crosses • Perpendicular lines—same plane, crosses to form a right angle

  4. Angles from lines cut by transversals • Exterior angles • Interior angles • Consecutive interior angles • Alternate interior angles • Alternate exterior • Corresponding angles 2 4 1 3 6 5 8 7

  5. Theorems: IF TWO PARALLEL LINES CUT BY A TRANSVERSAL-- • Each pair of corresponding angles is congruent (corr /_‘s ~ ) = • Each pair of alternate interior angles is congruent (AIA) • Each pair of consecutive interior angles is supplementary (CIA) • Each pair of alternate exterior angles is congruent (AEA)

  6. Slopes: Parallel & Perpendicular Lines -Given (x1, y1) & (x2, y2): m = ,x1≠ x2 -Two nonvertical lines have the same slope if and only if they are parallel. m1 = m2 iff ℓ1 // ℓ2 -Two nonvertical lines are perpendicular if and only if the products of their slopes is –1 ℓ1┴ ℓ2 iff m1·m2 = –1 ** iff means if and only if: ifm1 = m2ℓ1 // ℓ2AND ifℓ1 // ℓ2 m1 = m2

  7. If there is a line & a point not one the line, only one line can go through the point & be parallel to the line … then the lines are parallel If you want to prove lines are parallel match the IF If2 co-planar lines are cut by a transversal that makes: -the corresponding angles congruent… -or alt.exterior angles congruent… -or a pair of consec.interior angles congruent… -or a pair of alt.interior angles are congruent… orIf 2 co-planar lines are perpendicular to the sameline…

  8. Distance Definition: The distance between a line and a point not on the line = the length of the perpendicular segment that joins the point to the line Definition: the distance between two parallel lines = the length of the perpendicular segment that joins the two line (90°angles) If a line is perpendicular to one of two parallel line, then it is perpendicular to both. • distance

  9. 1 Given 2 Corresponding 3 Transitivity 4 Transitivity 5 J K A EXAMPLE: Proof 1 2 3 4 Given: , , Prove: O N 1 , 2 3 4 5

  10. Web Resources http://math.about.com/gi/dynamic/offsite.htm?zi=1/XJ/Ya&sdn=math&cdn=education&tm=29&gps=87_9_796_428&f=00&tt=14&bt=1&bts=0&zu=http%3A//library.thinkquest.org/C0110248/geometry/analytic.htm http://hotmath.com/?referrer=goo-cg-geo&gclid=COOiwIXV7IwCFRE4OAodJVev6g http://www.math.psu.edu/geom/koltsova/index.html

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