1. 3.6 Prove Theorems About Perpendicular Lines

2. Objectives Recognize relationships within  lines Prove that two lines are parallel based on given  information

3. Theorems Theorem 3.8 If 2 lines intersect to form a linear pair of   s, then the lines are . Theorem 3.9 If 2 lines are , then they intersect to form 4 right  s. Theorem 3.10 If 2 sides of 2 adjacent acute  s are , then the  s are complementary.

4. EXAMPLE 1 Draw Conclusions In the diagram, AB BC. What can you conclude about 1 and 2 ? SOLUTION AB and BC are perpendicular, so by Theorem 3.9, they form four right angles. You can conclude that 1 and 2 are right angles, so 1  2.

5. EXAMPLE 2 Prove Theorem 3.10 Prove that if two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Given ED EF Prove 7 and 8 are complementary.

6. YOUR TURN Given that ABC  ABD, what can you conclude about 3 and 4 ? Explain how you know. 1. They are complementary. Sample Answer: ABD is a right angle since 2 lines intersect to form a linear pair of congruent angles (Theorem 3.8 ), 3 and 4 are complementary. ANSWER

7. Theorems Theorem 3.11 (  Transversal Theorem) If a transversal is  to one or two || lines, then it is  to the other. Theorem 3.12 (Lines  to a Transversal Theorem) In a plane, if 2 lines are  to the same line, then they are || to each other.

8. EXAMPLE 3 Draw Conclusions SOLUTION Lines p and q are both perpendicular to s, so by Theorem 3.12, p || q. Also, lines s and t are both perpendicular to q, so by Theroem 3.12, s || t. Determine which lines, if any, must be parallel in the diagram. Explain your reasoning.

9. YOUR TURN Use the diagram at the right. 3. Is b || a ? Explain your reasoning. 4. Is b c ? Explain your reasoning. 3. yes; Lines Perpendicular to a Transversal Theorem. 4. yes; c || d by the Lines Perpendicular to a Transversal Theorem, therefore b c by the Perpendicular Transversal Theorem. ANSWER

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