Geometry Chapter 3 Parallel and Perpendicular Lines Lesson 1 Lines and Angles Parallel lines: Perpendicular lines: Skew lines: Parallel Planes: Examples: Use the diagram to identify the following: 1a. a pair of parallel segments 1b. a pair of skew segments 1c. a pair of perpendicular segments 1d. a pair of parallel planes 1e. a pair of perpendicular planes Page 1
Give an example of each angle pair. 2a. corresponding angles 2b. alternate interior angles 2c. alternate exterior angles 2d. same-side interior angles 2e. same-side exterior angles 2f. vertical angles Use the diagram to identify the transversal and classify each angle pair. 3a. 1 and 3 3b. 2 and 6 3c. 4 and 6 Lesson 2 Angles formed by Parallel Lines and Transversals Theorem Corresponding Angle Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. Alternate Interior Angle Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Alternate Exterior Angle Theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Page 2
Same-Side Interior Angle Theorem If two parallel lines are cut by a transversal, then same-side interior angles are supplementary. Same-Side Exterior Angle Theorem If two parallel lines are cut by a transversal, then same-side exterior angles are supplementary. Examples: 1. Find x and each angle measure. 2. Find x and each angle measure. 2b. Find x 3. Find x and y in the diagram. m 1= 2x+27 m 2= 2x + y Page 3
Lesson 3 Proving Lines Parallel Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem. Examples: 1. Use the diagram at the right, each set of given information, and the theorems that you have learned to show that l // m a. 4 8 b. 1 7 c. m 3 = (4x 80), m 7 = (3x 50), x = 30 d. m 3 = (10x + 8), m 5 (25x 3), x = 5 2. A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m 1= (8x + 20) and m 2 = (2x + 10). If x = 15, show that pieces A and B are parallel. Given: p // r, 1 3 Prove: l // m Statements 1. Reasons 2. 3. 4. Page 4
Given: 1 4 and 3 and 4 are supplementary Prove: t // m Statements Reasons 1. 2. 3. 4. 5. 6. 7. Lesson 4 Perpendicular Lines Perpendicular Bisector: Distance from a point to a line: Page 5
Examples: 1a. Name the shortest segment from Point A to line BC. 1b. Write and solve the inequality for x. 2. A carpenter s square forms a right angle. A carpenter places the square so that one side is parallel to an edge of a board, and then draws a line along the other side of the square. Then he slides the square to the right and draws a second line. Why must the two lines be parallel? Given: r // s, 1 2 Prove: r t Statements 1. Reasons 2. 3. 4. Lesson 5 Slopes of Lines The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope. Rise: Run: Slope: Page 6
Examples: 1. Use the slope formula to determine the slope of each line. a. b b. c. d. Examples: 2. Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin s distance from home at a given time. Find and interpret the slope of the line. Parallel Line Theorem: In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. Ex: Perpendicular Line Theorem: In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. Ex: Page 7
3. Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither. a. b. c. Lesson 6 Lines in the Coordinate Plane The equation of a line can be written in many different forms. The point-slope and slope-intercept forms of a line are equivalent. Because the slope of a vertical line is undefined, these forms cannot be used to write the equation of a vertical line. Page 8
Examples: 1. Write the equation of each line in the given form. a. the line with slope 6 through (3, 4) in point-slope form 2. Graph each line. a. y = ½ x + 1 b. the line through ( 1, 0) and (1, 2) in slope-intercept form b. y 3 = -2(x + 4) c. the line with the x-intercept 3 and y-intercept 5 in point slope form c. y = -3 d. y 1 = - 2 3 (x + 2) A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Lines that coincide are the same line, but the equations may be written in different forms. 3. Determine whether the lines are parallel, intersect, or coincide. a. y = 3x + 7, y = 3x 4 c. 2y 4x = 16, y 10 = 2(x - 1) b. d. 3x + 5y = 2, 3x + 6 = -5y 4. Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same? Page 9