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Perpendicular Lines

This resource explores the concept of perpendicular lines, including their definitions, properties, and theorems related to angles and distance from a point to a line. Key topics include the perpendicular bisector of a segment, perpendicular transversal theorem, and the reasoning behind the relationship between coplanar lines and their parallelism. It provides clear examples, proofs, and inequalities to enhance understanding of these geometric principles, making it ideal for students studying geometry.

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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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