1. What is the distance between the points (2, 3) and (5, 7)?

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Presentation transcript:

1. What is the distance between the points (2, 3) and (5, 7)? ANSWER 5 2. If m DBC = 90°, what is m ABD? ANSWER 90°

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.

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.

GUIDED PRACTICE for Examples 1 and 2 Given that ABC  ABD, what can you conclude about 3 and 4? Explain how you know. 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

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. 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

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?

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 20 50 – 30 3 2 . The segment SR has a slope of 120 – 110 = 10 15 35 – 50 – 2 3 . The segment SR is perpendicular to the leg so the distance SR is (35 – 50)2 + (120 – 110)2 18.0 inches. d = 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? 5. about 1.3 6. about 2.2 ANSWER

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. (2, 3); 2.8 ANSWER

Daily Homework Quiz 1. Find m 3. 18° ANSWER 2. How do you know that a and b are parallel? Both are perpendicular to c. ANSWER

Daily Homework Quiz 3. Find the distance between the two parallel lines. Round to the nearest tenth. 6.4 ANSWER