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Warm-Up, 1

Warm-Up, 1. What is the distance from point P to line l ?. Warm-Up, 2. What is the shortest distance from P to line l ?. Distance from a Point to a Line.

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Warm-Up, 1

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  1. Warm-Up, 1 What is the distance from point P to line l?

  2. Warm-Up, 2 What is the shortest distance from P to line l?

  3. Distance from a Point to a Line The distance from a point to a line is the length of the perpendicular segment from the point to the line. This is the shortest distance from the point to the line.

  4. Distance Between Parallel Lines Likewise, the distance between two parallel lines is the length of any perpendicular segment joining the two lines.

  5. 3.6 Prove Theorems About Perpendicular Lines Objectives: • To find the distance from a point to a line • To construct perpendicular lines with a compass and straightedge • To prove theorems about perpendicular lines

  6. Perpendicular Postulate If we wanted to measure the distance between a point and a line, we can employ the Perpendicular Postulate, a compass, and a straightedge, since there exists only one perpendicular line from a point to a line.

  7. Perpendicular: Point Not on Line • Draw a line and point P not on the line.

  8. Perpendicular: Point Not on Line • Using P as center, draw an arc that intersects the line twice. Label the points of intersection A and B.

  9. Perpendicular: Point Not on Line • Using the same compass setting, draw two intersecting arcs below the line, one centered at A, the other centered at B. Label the point of intersection Q.

  10. Perpendicular: Point Not on Line • Draw a line through points P and Q.

  11. Perpendicular Off: Video Click on the button to watch a video of the construction.

  12. While We’re At It… The previous construction assumed that you were drawing a perpendicular through a point not on a line. What if the point was actually on the line? We can still use our compass and straightedge to draw the perpendicular line.

  13. Perpendicular: Point On Line • Draw a line and a point P on the line.

  14. Perpendicular: Point On Line • Using P as center, draw an arc below the line that intersects the line twice. Label the points of intersection A and B.

  15. Perpendicular: Point On Line • Using the same compass setting (but larger than the first one), draw two intersecting arcs below the line, one centered at A, the other centered at B. Label the point of intersection Q.

  16. Perpendicular: Point On Line • Draw a line from through P and Q.

  17. Perpendicular On: Video Click on the button to watch a video of the construction.

  18. Theorems Galore! If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. • Proof Hints: • Write an equation based on the angles forming a linear pair. • Do some substitution and solve for one of the angles. It should be 90°.

  19. Theorems Galore! If two lines are perpendicular, then they intersect to form four right angles. • Proof Hints: • Use definition of perpendicular lines to find one right angle. • Use vertical and linear pairs of angles to find three more right angles.

  20. Theorems Galore! If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. • Proof Hints: • Use definition of perpendicular to get the measure of ABC. • Use Angle Addition Postulate, Substitution, and Definition of complementary angles to finish the proof.

  21. Theorems Galore! Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. • Proof Hints: • Use definition of perpendicular lines to find one right angle. • Use Corresponding Angles Postulate to find a right angle on the other line. j

  22. Theorems Galore! Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. • Proof Hints: • Use definition of perpendicular lines to find a right angle on each parallel line. • Use Converse of Corresponding Angles Postulate to prove the lines are parallel.

  23. Parallel Lines the Right Way Finally, we can use the Lines Perpendicular to a Transversal Theorem to construct a couple of parallel lines.

  24. Parallel Lines the Right Way • Construct a line with points A and B.

  25. Parallel Lines the Right Way • Using the same compass setting construct two arcs, one centered at A and one centered at B.

  26. Parallel Lines the Right Way • Construct points of intersection C, D, E, and F.

  27. Parallel Lines the Right Way • Using a slightly larger compass setting, construct two sets of intersecting arcs: one set centered at C and D; the other centered at E and F.

  28. Parallel Lines the Right Way • Construct points of intersection G and H. • Draw a line through A and G. • Draw a line through B and H.

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