Chapter 3: Parallel and Perpendicular

Lines

Lesson 1: Parallel Lines and Transversals

Definitions Parallel lines ( || )- coplanar lines that do

not intersect (arrows on lines indicate which sets are parallel to each other)

Parallel planes- two or more planes that do not intersect

Skew lines- lines that do not intersect but are not parallel (are not coplanar)

Transversal- a line that intersects two or more lines in a plane at different points

Frayer Model

Alternate Exterior Angles Alternate Interior Angles

Corresponding Angles Consecutive Interior Angles

Pairs of angles formed by parallel lines and a transversal (see graphic organizer for examples)

Exterior angles: outside the two parallel lines Interior angles: between the two parallel lines Consecutive Interior angles: between the two

parallel lines, on the same side of the transversal Alternate Exterior angles: outside the two

parallel lines, on different sides of the transversal Alternate Interior angles: between the two

parallel lines, on different sides of the transversal Corresponding angles: one outside the parallel

lines, one inside the parallel lines and both on the same side of the transversal

A. Name all segments parallel to BC.B. Name a segment skew to EH.

C. Name a plane parallel to plane ABG.

Classify the relationship between each set of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles

A. 2 and 6

B. 1 and 7

C. 3 and 8

D. 3 and 5

A. A group of nature trails is shown. Identify the sets of lines to which line a is a transversal.B. A group of nature trails is shown. Identify the sets of lines to which line b is a transversal.C. A group of nature trails is shown. Identify the sets of lines to which line c is a transversal.

Chapter 3: Parallel and Perpendicular

Lines

Lesson 2: Angles and Parallel Lines

If two parallel lines are cut by a transversal, then… (see graphic organizer)

the alternate interior angles are congruent the consecutive interior angles are

supplementary the alternate exterior angles are

congruent the corresponding angles are congruent

In a plane, if a line is perpendicular to one of the two parallel lines, then it is also perpendicular to the other line.

A. In the figure, m11 = 51. Find m15. Tell which postulates (or theorems) you used.

B. In the figure, m11 = 51. Find m16. Tell which postulates (or theorems) you used.

A. In the figure, a || b and m18 = 42. Find m22.

B. In the figure, a || b and m18 = 42. Find m25.

A. ALGEBRA If m5 = 2x – 10, and m7 = x + 15, find x.

B. ALGEBRA If m4 = 4(y – 25), and m8 = 4y, find y.

A. ALGEBRA If m1 = 9x + 6, m2 = 2(5x – 3), and m3 = 5y + 14, find

x. B. ALGEBRA If m1 = 9x + 6, m2 = 2(5x – 3), and

m3 = 5y + 14, find y.

Chapter 3: Parallel and Perpendicular

Lines

Lesson 3: Slopes of Lines

Slope The ratio of the vertical rise over the

horizontal run Can be used to describe a rate of change

Two non-vertical lines have the same slope if and only if they are parallel

Two non-vertical lines are perpendicular if and only if the product of their slopes is -1

Foldable Step 1: fold the paper into 3 columns/sections

Step 2: fold the top edge down about ½ inch to form a place for titles. Unfold the paper and turn it vertically.

Step 3: title the top row “Slope”, the middle row “Slope-intercept form” and the bottom row “Point-slope form”

Slope

Rise = 0 zero slope (horizontal line)Run = 0 undefined (vertical line)

Parallel = same slope Perpendicular = one slope is the reciprocal and opposite sign

of the other

Ex: find the slope of a line containing (4, 6) and (-2, 8)

12

12

xx

yym

Find the slope of the line.

Find the slope of the line.

Find the slope of the line.

Find the slope of the line.

Determine whether FG and HJ are parallel, perpendicular, or neither for F(1, –3), G(–2, –1), H(5, 0), and J(6, 3).

(DO NOT GRAPH TO FIGURE THIS OUT!!)

Determine whether AB and CD are parallel, perpendicular, or neither for A(–2, –1),

B(4, 5), C(6, 1), and D(9, –2)

A. Graph the line that contains Q(5, 1) and is parallel to MN with M(–2, 4) and N(2, 1).

B. Graph the line that contains (-1, -3) and is perpendicular to MN for M(–3, 4) and N(5, –8)?

Chapter 3: Parallel and Perpendicular

Lines

Lesson 4: Equations of Lines

Slope-intercept form : y = mx + b

Slope and y-intercept

Two ordered-pairs (one is y-intercept)

Two ordered-pairs (neither is y-intercept)

m = -4y-intercept = 7

(4, 1) (0, -2) (3, 3) (2, 0)

* This should be your middle row on the foldable

Point-slope form: )( 11 xxmyy

Slope and one ordered-pair Two ordered-pairs

m = (7, 2) (8, -2) (-3, -1)

3

1

* This should be your bottom row on the foldable

Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of –3.

Write the equation in slope-intercept form and thenWrite an equation in point-slope form of the line

whose slope is that contains (–10, 8). Then

graph the line.

Write an equation in slope-intercept form for a line containing (4, 9) and (–2, 0).

Write an equation in point-slope form for a line containing (–3, –7) and (–1, 3).

Chapter 3: Parallel and Perpendicular

Lines

Lesson 5: Proving Lines Parallel

Two lines are parallel if they are cut by a transversal so that… (see graphic organizer)

Corresponding angles are congruent Alternate exterior angles are congruent Consecutive interior angles are supplementary Alternate interior angles are congruent They are both perpendicular to the transversal

If given a line and a point not on the line, there is exactly one line through that point that is parallel to the given line

A. Given 1 3, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer.

B. Given m1 = 103 and m4 = 100, is it possible to prove that any of the lines shown are parallel? If so, state the postulate or theorem that justifies your answer.

Find ZYN so that || . Show your work.

A. Given 9 13, which segments are parallel?

B. Given 2 5, which segments are parallel?

C. Find x so that AB || HI if m1 = 4x + 6 and

m14 = 7x – 27. _____

Perpendicular Lines and Distance The shortest distance between a line and a point

not on the line is the length of the perpendicular line connecting them

Equidistant: the same distance- parallel lines are equidistant because they never get any closer or farther apart

The distance between two parallel lines is the distance between one line and any point on the other line

In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other