Perpendicular and Parallel Lines Chapter 3 Perpendicular and Parallel Lines

3.1 – Lines and Angles Two lines are PARALLEL LINES if they are coplanar and they do not intersect. Two lines are SKEW LINES if they are not coplanar and they do not intersect. Two planes that do not intersect are called PARALLEL PLANES.

Parallel Postulates If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. P

Transversals and Angles A TRANSVERSAL is a line that intersects two or more coplanar lines at different points.

Corresponding Angles Two angles are corresponding angles if they occupy corresponding positions. 1 2 3 4 5 6 8 7

Alternate Interior Angles Two angles are alternate interior angles if they lie between the two lines on opposite sides of the transversal. 1 2 3 4 5 6 8 7

Alternate Exterior Angles Two angles are alternate exterior angles if they lie outside the two lines on opposite sides of the transversal. 1 2 3 4 5 6 8 7

Consecutive Interior Angles Two angles are consecutive int. angles if they lie between the two lines on the same side of the transversal. (aka: same side interior angles) 1 2 3 4 5 6 8 7

Parallel, skew, or perpendicular UT and WT are: _________ RS and VW are: ___________ TW and WX are: ___________ R S U T X V W

Lines and Planes Name a line parallel to HJ. Name a line skew to GH. Name a line perpendicular to JH. M L K J N G H

Angles 6 and 10 are __________ angles. 12 and 6 are __________ angles. 5 7 8 10 9 11 12

3.2 Perpendicular Lines If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. g h g h

Perpendicular Lines If two sides of two adjacent angles are perpendicular, then the angles are complementary.

Perpendicular Lines If two lines are perpendicular, then they intersect to form four right angles.

3.3 Parallel Lines and Transversals Corresponding Angles Postulate If 2 parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2

Alternate Interior Angles If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 5 3 4 6

Consecutive Interior Angles If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 5 3 6 4

Alternate Exterior Angles If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 5 3 4 6

Perpendicular Transversal If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Examples Find m 1 and m 2. 1 2 105

Examples Find m 1 and m 2. 1 2 135

Examples Find x. 64 2x

Examples Find x. (3x – 30)

3.4 Parallel Lines 3.4 is basically the converse of 3.3.

Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. 1 2

Alternate Interior Angles Converse If two parallel lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. 3 4

Consecutive Interior Angles Converse If two parallel lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. 5 6

Alternate Exterior Angles Converse If two parallel lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. 5 6

3.5 Properties of Parallel Lines If two lines are parallel to the same line, then they are parallel to each other. p q r

Properties of Parallel Lines In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. p m n

Proving Lines are Parallel State which pair of lines are parallel and why. m A = m 6 B D F 5 6 A C E

Proving Lines are Parallel m 6 = m 8 B D 8 7 5 4 6 A C

Proving Lines are Parallel 1 = 90 , 5 = 90 Y Z 1 2 4 5 X W

Which lines are parallel if given: ~ 1 = 4 6 = 4 2 + 3 = 5 1 = 7 1 = 8 2 + 3 + 8 = 180 ~ l m t ~ j ~ 8 6 ~ 2 1 4 5 3 k 7

Find x and y. Given: AB II EF , AE II BF A B C (2x-7) (3x-34) (4y+1) D

3.6-3.7 Coordinate Geometry Slope of a line: The equation of a line: Slope-intercept form: y = mx + b Point-slope form: y – y1 = m (x – x1)

Parallel and Perpendicular Slopes In a coordinate plane, two nonvertical lines are parallel iff they have the same slope. (Any two vertical lines are parallel.) In a coordinate plane, two nonvertical lines are perpendicular iff the product of their slopes is -1. (Vertical and horizontal lines are perpendicular.)

Parallel and Perpendicular Slopes Parallel lines have the same slopes. Perpendicular lines have slopes that are OPPOSITE RECIPROCALS. (Line 1 has a slope of -2. Line 2 is perpendicular to Line 1 and has a slope of ½ .

Horizontal Lines Horizontal lines have a slope = 0 (zero) The equation of a horizontal line is y = b

Vertical Lines Vertical lines have undefined slope (und) The equation of a vertical line is x = a

Finding the equation of a line Write the equation of the line that goes through (1,1) with a slope of 2.

Finding the equation of a line Find the slope of the line between the points (0,6) and (5,2). Then write the equation of the line.

Parallel? Line 1 passes through (0,6) and (2,0).

Perpendicular? Line 1 passes through (4,2) and (1,-4).

3.6-3.7 Writing equations Write the equation of the line which passes through point (4,9) and has a slope of -2.

Equations of lines Write the equation of the line parallel to the line y = -2/5 x + 3 and passing through the point (-5,0).

Equations of lines Write the equation of the line parallel to the line 3x + 2y = 4 and passing through the point (-4, 5).

Equations of lines Write the equation of the line parallel to the x-axis and passing through the point (3,-6).

Equations of lines Write the equation of the line parallel to the y-axis and passing through the point (4,8).

Parallel Lines? Line p1 passes through (0,-3) & (1,-2). Line p2 passes through (5,4) & (-4,-4). Line p3 passes through (-6,-1) & (3,7). Find the slope of each line. Which lines are parallel?

Slope Find the slope of the line passing through the given points.

Parallel? Determine if the two lines are parallel.

Slope What is the slope of the line?

Slope Find the slope of the line that passes through the given points.

Parallel? Determine if the two lines are parallel.

Perpendicular? With the given slopes, are the lines perpendicular? m1 = 2 ; m2 = - ½ m1 = -1 ; m2 = 1 m1 = 5/7 ; m2 = - 7/5

Perpendicular? Find the slopes of the two lines. Determine if they are perpendicular.

Perpendicular? Line 1: y = 3x ; Line 2: y = (-1/3)x – 2

Perpendicular? Line 1: 3y + 2x = -36 ; Line 2: 4y – 3x = 16

Writing equations Write the equation of the line perpendicular to the line y = (-3/4)x + 6 that goes through the point (8,0)

Writing Equations Write the equation of the line that goes through point (-3,-4) and that is perpendicular to y = 3x + 5.

Perpendicular, Parallel or Neither y = -2x – 1 y = -2x – 3 y = 4x + 10 y = -2x + 5