3.6 Prove Theorems About Perpendicular Lines

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 3.9If 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. In the diagram, ABBC. What can you conclude about 1 and 2? SOLUTION ABand 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. EXAMPLE 1 Draw Conclusions

5. Prove that if two sides of two adjacentacute angles are perpendicular, then theangles are complementary. GivenED EF Prove7and8 are complementary. EXAMPLE 2 Prove Theorem 3.10

6. Given that ABC ABD, what can you conclude about 3 and 4? Explain how you know. ANSWER 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. YOUR TURN

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. Determine which lines, if any, must be parallel in the diagram. Explain your reasoning. 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. EXAMPLE 3 Draw Conclusions

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

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. 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? EXAMPLE 4 Find the distance between two parallel lines

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

14. 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? ANSWER 5. about 1.3 6.about 2.2 YOUR TURN

15. ANSWER (2, 3); 2.8 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.

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