Chapter 3.6 Notes: Prove Theorems about Perpendicular Lines Goal: You will find the distance between a point and a line.

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Presentation transcript:

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,. What can you conclude about and. 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, what can you conclude about and ? Explain how you know. 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 ? Explain your reasoning. 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. 2x = 90 2x + 54 = 90 2x = 36 x = 18°

1. 3x – = 90 3x + 27 = 90 3x = 63 x = 21° 38°

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?

a Ex.7: In the figure, and are congruent. What can you conclude about ? a 1 b 2

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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