Chapter 3: Parallel and Perpendicular Lines
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Transcript of Chapter 3: Parallel and Perpendicular Lines
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Chapter 3: Parallel and Perpendicular
Lines
Lesson 1: Parallel Lines and Transversals
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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
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Frayer Model
Alternate Exterior Angles Alternate Interior Angles
Corresponding Angles Consecutive Interior Angles
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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
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A. Name all segments parallel to BC.B. Name a segment skew to EH.
C. Name a plane parallel to plane ABG.
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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
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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.
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Chapter 3: Parallel and Perpendicular
Lines
Lesson 2: Angles and Parallel Lines
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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.
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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.
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A. In the figure, a || b and m18 = 42. Find m22.
B. In the figure, a || b and m18 = 42. Find m25.
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A. ALGEBRA If m5 = 2x – 10, and m7 = x + 15, find x.
B. ALGEBRA If m4 = 4(y – 25), and m8 = 4y, find y.
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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.
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Chapter 3: Parallel and Perpendicular
Lines
Lesson 3: Slopes of Lines
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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
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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”
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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
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Find the slope of the line.
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Find the slope of the line.
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Find the slope of the line.
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Find the slope of the line.
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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!!)
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Determine whether AB and CD are parallel, perpendicular, or neither for A(–2, –1),
B(4, 5), C(6, 1), and D(9, –2)
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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)?
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Chapter 3: Parallel and Perpendicular
Lines
Lesson 4: Equations of Lines
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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
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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
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Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of –3.
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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.
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Write an equation in slope-intercept form for a line containing (4, 9) and (–2, 0).
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Write an equation in point-slope form for a line containing (–3, –7) and (–1, 3).
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Chapter 3: Parallel and Perpendicular
Lines
Lesson 5: Proving Lines Parallel
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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
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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.
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Find ZYN so that || . Show your work.
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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. _____
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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