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Perpendicular Lines

Perpendicular Lines. Geometry (Holt 3-4) K.Santos. Perpendicular Bisector. Perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint. (could be a segment or ray) s M t Line s is perpendicular to line t at it’s midpoint M .

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Perpendicular Lines

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  1. Perpendicular Lines Geometry (Holt 3-4) K.Santos

  2. Perpendicular Bisector Perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint. (could be a segment or ray) s M t Line s is perpendicular to line t at it’s midpoint M

  3. Distance from a point to a line The shortest segment from a point to a line is perpendicular to the line. Distance form a point to a line is the length of the perpendicular segment from the point to the line.

  4. Theorem (3-4-1) If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. Given: <1 and < 2 are a linear pair n Then: m ⊥ n 1 2 m

  5. Perpendicular Transversal Theorem (3-4-2) In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line. a b Given: a||b t┴a t Then: t ┴b

  6. Theorem (3-4-3) Theorem: If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. Given: a┴ t a b┴t Then: a||b b t

  7. Proof of previous theorem Given: a ┴ t b ┴t 1 a Prove: a||b 2 b t Statements Reasons 1. a ┴ t, b ┴t 1. Given 2. < 1 and < 2 are right angles 2. Definition of perpendicular lines 3. <1 <2 3. all right angles are congruent 4. a||b 4. If corresponding angles are congruent then the lines are parallel

  8. Theorem If two coplanar lines are parallel to the same line, then they are parallel to each other. t Given: a ||b a b ||c b Then: a || c c

  9. Example: Use the picture at the right to answer the questions below: C P B x – 8 12 Name the shortest segment from point A to . Write and solve an inequality for x. AC > AP x – 8 > 12 x > 20

  10. Example Given the information below what can you conclude about lines a and d? a ||b a b┴c c||db a ___ d? cd Draw a picture with all the line in it and then make a conclusion about lines a and d. a ┴ d

  11. Proof Given: r||s t <1 <2 1 3 r Prove: rs 2 s Statements Reasons 1. r||s 1. given 2. <2 <3 2. Corresponding Angles postulate 3. <1 <2 3. given 4. <1 <3 4. Transitive Property (2, 3) 5. rs 5. If two intersecting lines form a then the lines are perpendicular

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