# Chapter 3 - PowerPoint PPT Presentation

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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.

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Chapter 3

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## Chapter 3

Perpendicular and Parallel Lines

### 3.1 – Lines and Angles

• Two lines are PARALLEL LINESif 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.

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### Alternate Interior Angles

• Two angles are alternate interior angles if they lie between the two lines on opposite sides of the transversal.

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### Alternate Exterior Angles

• Two angles are alternate exterior angles if they lie outside the two lines on opposite sides of the transversal.

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### 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)

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### 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.

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K

J

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H

### Angles

• 6 and 10 are __________ angles.

• 12 and 6 are __________ angles.

• 7 and 9 are __________ angles.

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### 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.

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2

### Alternate Interior Angles

• If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

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### Consecutive Interior Angles

• If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

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### Alternate Exterior Angles

• If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

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### 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.

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2

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### Examples

• Find m 1 and m 2.

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2

135

• Find x.

64

2x

• 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.

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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.

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### 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.

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### 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.

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### 3.5 Properties of Parallel Lines

• If two lines are parallel to the same line, then they are parallel to each other.

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### Properties of Parallel Lines

• In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

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### Proving Lines are Parallel

• State which pair of lines are parallel and why.

• m A = m 6

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F

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6

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• m 6 = m 8

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C

### Proving Lines are Parallel

• 1 = 90 , 5 = 90

Y

Z

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5

X

W

1 = 4

6 = 4

2 + 3 = 5

1 = 7

1 = 8

2 + 3 + 8 = 180

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### Find x and y.

• Given: AB II EF , AE II BF

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(2x-7)

(3x-34)

(4y+1)

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E

### 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).

• Line 2 passes through (-2,6) and (0,1).

• Line 3 passes through (-6,5) and (-4,0)

### Perpendicular?

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

• Line 2 passes through (-1,2) and (5,-1).

### 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

• Line 1: y = (1/3)x – 10 ; Line 2: y = 3x

### Perpendicular?

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

• Line 1: 3y – 4x = 3 ; Line 2: 4y + 3x = -12

### 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.

y = -2x – 1

y = -2x – 3

y = 4x + 10

y = -2x + 5