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3.6 Prove Theorems About Perpendicular Lines Objective: Find the distance between a point and a line What can you conclude if… Theorem 3.8 • If 2 lines intersect to form a linear pair of congruent angles, then the lines must be perpendicular. What can you conclude if… Theorem 3.9 • If 2 lines are ┴ then they form 4 congruent angles. 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. GUIDED PRACTICE for Examples 1 and 2 1. 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. EXAMPLE 2 Prove Theorem 3.10 Prove that if two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Given Prove ED EF 7 and 8 are complementary. What can you conclude if… Theorem 3.11 • Perpendicular Transversal Theorem: If a transversal is perpendicular to one of 2 parallel lines, then it’s perpendicular to both of them. What can you conclude if… Theorem 3.12 • Lines Perpendicular to a Transversal Theorem: If 2 lines are perpendicular to the same line, then they are parallel to each other. EXAMPLE 3 Draw Conclusions 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. GUIDED PRACTICE for Example 3 Use the diagram at the right. 3. Is b || a? Explain your reasoning. 4. Is b c? Explain your reasoning. ANSWER 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. Distance From a Point to a Line • Length of the perpendicular segment from the point to a line 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. Using the points P(30, 80) and R(50, 110), the slope of each leg is 110 – 80 = 30 = 3 . 2 20 50 – 30 The segment SR has a slope of 120 – 110 = – 10 = – 2 . 3 15 35 – 50 The segment SR is perpendicular to the leg so the distance SR is d= (35 – 50)2 + (120 – 110)2 18.0 inches. The length of SR is about 18.0 inches. GUIDED PRACTICE for Example 4 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 GUIDED PRACTICE for Example 4 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. ANSWER (2, 3); 2.8 Daily Homework Quiz 1. Find m ANSWER For use after Lesson 3.6 3. 18° 2. How do you know that a and b are parallel? ANSWER Both are perpendicular to c. Daily Homework Quiz For use after Lesson 3.6 3. Find the distance between the two parallel lines. Round to the nearest tenth. ANSWER 6.4 Homework • 1 – 27, 29 – 31 • Bonus: 28, 35 – 38