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Parallel Lines and Transversals

Angles and Parallel Lines Distance

Equations of Lines

Proving Lines are Parallel

Parallel Lines and Transversals for $100

Define: Skew lines

Answer

Skew Lines - Lines that are not coplanar and do not intersect

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Parallel Lines and Transversals for $200

Define: Parallel Lines

Answer

Parallel Lines – Lines that are coplanar and do not intersect.

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Parallel Lines and Transversals for $300

Define: Transversal

Answer

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

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Parallel Lines and Transversals for $400

Name all the line segments parallel to AB

Answer

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CD, GH, EF

Parallel Lines and Transversals for $500

Name all of the line segments perpendicular to GC

Answer

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EG, GH, CA, CD

Angles and Parallel Lines for $100

Identify two pairs of consecutive interior angles in the following drawing given l || m:

l m

n1 4

2 3

5 8

6 7

Answer

<4 and <5, <3 and <6Note: <4 + <5 = 180 degrees

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l m

n1 4

2 3

5 8

6 7

Angles and Parallel Lines for $200

Identify two pairs of corresponding angles in the following drawing given l || m:

l m

n1 4

2 3

5 8

6 7

Answer

1 and 5, 4 and 8, 2 and 6, 3 and 7

NOTE: 1 5

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l m

n1 4

2 3

5 8

6 7

Angles and Parallel Lines for $300

Identify two pairs of alternate interior angles in the following drawing given l || m:

l m

n1 4

2 3

5 8

6 7

Answer

<4 and <6, <3 and <5

Note: <4 <6

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l m

n1 4

2 3

5 8

6 7

Angles and Parallel Lines for $400

Given r is parallel to t, find the measure of angle 6

Answer

Back

<2 = 135 degree angle – corresponding angles.

<2 and < 6 are supplementary

135 + < 6 = 180

<6 = 45 degrees

Angles and Parallel Lines for $500

m<1 = 6x, and m<3 = 7x - 20. Find the value of x for p to be parallel to q. The diagram is not to scale.

Answer

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m<1 must be congruent to m<3 for p || q

6x = 7x – 20

20 = x

Equations of Lines for $100

Write the equation of the line in slope-intercept form:

The line with a slope of -5 through point (-2, -4)

AnswerPoint: (-2, -4)

m = -5

Slope-intercept Form:

y = mx + b

-4 = -5(-2) + b

-4 = 10 + b

-14 = b

Thus, y = -5x - 14

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Equations of Lines for $200

Write the equation of the line in slope-intercept form:

The line through points

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

Answer

Point: (-2, 3) Point: (0, -1)

m = (y2 – y1)/(x2 – x1)m = (-1- 3)/(0 – -2) = -4/2 = -2Slope-intercept Form:y = mx + b-1 = -2(0) + b-1 = bThus, y = -2x - 1

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Equations of Lines for $300

Write the equation of the line in point-slope form:

The line through points

(2, -3) and (-2, 3)

Answer

Points: (2, -3) and (-2, 3)

m = (y2 – y1)/(x2 – x1)m = (3- -3)/(-2 – 2) = 6/-4 = -3/2Point-Slope Form:

y – y1 = m(x – x1)

where (x1, y1) is a point on the lineThus, the equation of the line isy – 3 = -3/2(x - -2)y – 3 = -3/2(x + 2)

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Equations of Lines for $400

Write the equation of a line perpendicular to the given line that intersects the given line on the y-axis. Write your answer in point-slope form:

y = 3x - 8

Answery = 3x – 8So, m = 3, a point on the line = (0,-8)Point-Slope Form:

y – y1 = m(x – x1) y - -8 = 3(x – 0)y + 8 = 3xSlope of the Perpendicular line: (-1/3)y + 8 = (-1/3)x

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Equations of Lines for $500

Graph the following line:

y = 3x - 2

Answer

3x - 2

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Proving Lines to be Parallel for $100

Which 2 lines are parallel?

a) 5y = -3x - 5

b) 5y = -1 – 3x

c) 3y – 2x = -1

Answer

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Writing the lines in slope-intercept form:

a) 5y = -3x – 5

y = (-3/5)x – 1

b) 5y = -1 – 3x

y = (-3/5)x – (1/5)

c) 3y – 2x = -13y = 2x – 1y = (2/3)x – (1/3)

a||b

Proving Lines to be Parallel for $200

Given: <3 is supplemental to <8

Prove: p || r

Answer

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Statements Reasons<3 is supplemental to <8 Given

<3 + <8 = 180 Def. of Supplemental angles

<3 + <4 = 180 Def of Supplementary angles

<4 is congruent to <8 Theorem: Two angles supplementary to the same angle are congruent

<4 and <8 are corresponding angles

Definition of corresponding angles

p || r Theorem: If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel

Proving Lines to be Parallel for $300

Given: <1 is congruent to <5

Prove: p || r

Answer

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Statements Reasons<1 is congruent to <5 Given

<4 is congruent to <1 Vertical Angles<4 <1 and <1 <5 thus, <4 <5

Transitive property of angle congruence

Thus, p || r Theorem: If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel

Proving Lines to be Parallel for $400

Suppose you have four pieces of wood like those shown below. If b = 40 degrees can you construct a frame with opposite sides parallel? Explain.

Answer

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No, they are different transversals, so there is no theorem to prove the sides are congruent

Proving Lines to be Parallel for $500

Write a paragraph proof of this theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

Given: r is perpendicular to s, t is perpendicular to sProve: r || t

AnswerBy the definition of perpendicular, r ┴ s

implies m<2 = 90, and t ┴ s implies m<6 = 90. Line s is a transversal. <2 and <6 are corresponding angles. By the Converse of the Corresponding Angles Postulate, r || t.

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Distance for $100

Define: Distance between lines

Answer

Distance between lines: the shortest distance between the two lines

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Distance for $200

Given that two lines are equidistance from a third line, what can you conclude?

Answer

The two lines are parallel to each other

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Distance for $300

Define: equidistant

Answer

Equidistant: The distance between two lines measured along a perpendicular line to the lines is always the same.

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Distance for $400

What are the steps to find the distance between two parallel lines?

Answer

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1) Write both lines in slope-intercept form

2) Find the equation of a line perpendicular to the two parallel lines

3) Find the intersection of the perpendicular line with each of the given two lines

4) Find the distance between the two points

Distance for $500

Find the distance between the given parallel lines

y = 2x – 32x – y = -4

Answer

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d = √(9.8)

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