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MCC8.G5. Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence. a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically. b. Apply properties of angle pairs formed by parallel lines cut by a transversal. c. Understand the properties of the ratio of segments of parallel lines cut by one or more transversals.

Parallel Lines and

Transversals

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Lesson Essential Question:What results can you find when a transversal intersects parallel lines?

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Definition

Two lines are parallel if they do not intersect.

m n

q

p

p and q are not parallel

A

B

DC

What would you call two lines which do not intersect?

ParallelA solid arrow placed on two lines of a diagram indicate the lines are parallel.

The symbol || is used to indicate parallel lines.

AB || CD

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Transversal Definition: A line that intersects two or more lines at

different points.

When a transversal t intersects line n and m, angles of the following types are formed:

Exterior anglesInterior anglesAlternative exterior anglesCorresponding anglesVertical angles

tm

n

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We will be most concerned with transversals that cut parallel lines.

When a transversal cuts parallel lines, special pairs of angles are formed that are sometimes congruent and sometimes supplementary.

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Angles and Parallel Lines

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

1. Corresponding angles

2. Alternate interior angles

3. Alternate exterior angles

4. Vertical angles

Continued…..

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Vertical Angles & Linear PairVertical Angles:

Supplementary Angles: 1 4, 2 3, 5 8, 6 7

Two angles that are opposite angles. Vertical angles are congruent.

1 & 2 , 2 & 4 , 4 &3, 3 & 1,

5 & 6, 6 & 8, 8 & 7, 7 & 5

angles that form a line (sum = 180)

1 23 4

5 6

7 8

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Corresponding Angles & Consecutive Angles

Corresponding Angles: Two angles that are the same but in a different location.

2 6, 1 5, 3 7, 4 8

1 2

3 4

5 6

7 8

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Alternate Angles

Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair).

Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal.

3 6, 4 5

2 7, 1 8

1 2

3 4

5 6

7 8

TRY IT OUT

123 4

567 8

The m < 1 is 60 degrees.What is the m<3 ?

60 degrees

TRY IT OUT

6012060 120

6012060 120

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Example: If line AB is parallel to line CD and s is parallel to t, find the measure of all the angles when m< 1 = 100°. Justify your answers.

m<2=80° m<3=100° m<4=80°

m<5=100° m<6=80° m<7=100° m<8=80°

m<9=100° m<10=80° m<11=100° m<12=80°

m<13=100° m<14=80° m<15=100° m<16=80°

t

16 15

1413

12 11

109

8 7

65

34

21

s

DC

BA

TRY IT OUT

x + 102x + 20

What do you know about the angles?

Write the equation.

Solve for x.

SUPPLEMENTARY

2x + 20 + x + 10 = 180

3x + 30 = 1803x = 150x = 50

TRY IT OUT

2x - 60

3x - 120

What do you know about the angles?

Write the equation.

Solve for x.

ALTERNATE INTERIOR

3x - 120 = 2x - 60

x = 60Subtract 2x from both sides

Add 120 to both sides