1. Chapter 3 Parallel and Perpendicular Lines LEARNING OBJECTIVES RECOGNIZE RELATIONSHIPS WITHIN  LINES PROVE THAT TWO LINES ARE PARALLEL BASED ON GIVEN  INFORMATION LESSON 3.6: PROVE THEOREMS ABOUT PERPENDICULAR LINES

2. Theorem 3.8 If two lines intersect to from a linear pair of congruent angles, then the two lines are perpendicular What does linear pair mean? If they are congruent what measure must they be? Ex. If  1   2 then g  h 12 g h

3. Theorem 3.9 If two lines are perpendicular, then they intersect to form four right angles. If g  h then  1,  2,  3, and  4 are right angles 2 1 4 3 g h

5. GUIDED PRACTICE

6. Theorem 3.10 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. If BA  BC, then  1 and  2 are complementary B A C 1 2

7. Check Point Given that  ABC   ABD what can you conclude about  3 and  4? CDB A 4 3

8. Theorem 3.11 – Perpendicular Transversal If a transversal is perpendicular to 1 of 2 // lines, then it is perpendicular to the other.

9. Theorem 3.12 – Lines Perpendicular Transversal In a plane, if two lines are perpendicular to the same line, then they are // to each other.

10. Distance from a Point to a Line The distance from a line to a point not on the line is the length of the segment ┴ to the line from the point. l A

11. Distance Between Parallel Lines Two lines in a plane are || if they are equidistant everywhere. To verify if two lines are equidistant find the distance between the two || lines by calculating the distance between one of the lines and any point on the other line.

12. EXAMPLE 4 Find the distance between two parallel lines SCULPTURE: The sculpture on the right is drawn on a graph where units are measured in inches. What is the approximate length of SR, the depth of a seat?

13. EXAMPLE 4 Find the distance between two parallel lines SOLUTION You need to find the length of a perpendicular segment from a back leg to a front leg on one side of the chair. The length of SR is about 18.0 inches. The segment SR is perpendicular to the leg so the distance SR is (35 – 50) 2 + (120 – 110) 2 18.0 inches. d = The segment SR has a slope of 120 – 110 = 10 15 35 – 50 – = 2 – 3. Using the points P(30, 80) and R(50, 110), the slope of each leg is 110 – 80 = 30 20 50 – 30 = 3 2.

14. YOUR TURN Use the graph at the right for Exercises 5 and 6. 5. What is the distance from point A to line c? 6. What is the distance from line c to line d? 5. about 1.3 6. about 2.2 ANSWER

15. YOUR TURN 7. Graph the line y = x + 1. What point on the line is the shortest distance from the point (4, 1). What is the distance? Round to the nearest tenth. (2, 3); 2.8 ANSWER

16. Homework Assignment Pg. 194 – 197 #2 – 10, 13 – 24, 26, 31, 35 – 38