Chapter 3 6 Notes Prove Theorems about Perpendicular
Chapter 3. 6 Notes: Prove Theorems about Perpendicular Lines Goal: You will find the distance between a point and a line.
• Theorem 3. 8: If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. • Theorem 3. 9 If two lines are perpendicular, then they intersect to form four right angles.
Ex. 1: In the diagram below, conclude about and. . What can you • Theorem 3. 10: If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Ex. 2: Prove that if two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Given: Prove: and are complementary.
Ex. 3: Given that conclude about know. and , what can you ? Explain how you • Theorem 3. 11 Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
• Theorem 3. 12 Lines Perpendicular to a Transversal Theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Ex. 4: Determine which lines, if any, must be parallel in the diagram. Explain your reasoning.
Ex. 5: Use the diagram below. a. Is b. Is ? Explain your reasoning.
Distance form a Line • The distance from a point to a line is the length of the perpendicular segment from the point to the line. • This perpendicular segment is the shortest distance between the point and a line. • The distance between two parallel lines is the length of any perpendicular segment joining the two lines.
1. 2 x + 18 + 36 = 90 2 x + 54 = 90 2 x = 36 x = 18°
1. 3 x – 11 + 38 = 90 38° 3 x + 27 = 90 3 x = 63 x = 21°
Ex. 6: The sculpture below is drawn on a graph where units are measured in inches. What is the approximate length of , the depth of a seat?
Ex. 7: In the figure, and can you conclude about a 1 2 are congruent. What ? b
Ex. 8: Use the graph below for (a) and (b). a. What is the distance from point A to line c? b. What is the distance from line c to line d?
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