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Geometry
Unit 3 Notes Packet
3.1 Identify Pairs of Lines and Angles
3.2 Use Parallel Lines and Transversals
3.3 Prove Lines are Parallel
3.4 Find and Use Slopes of Lines
3.5 Write and Graph Equations of Lines
3.6 Prove Theorems about Perpendicular
Lines
3.1 Identify Pairs of Lines and Angles
Two lines are considered to be parallel lines if they are _______________ and do not
________________.
Two lines are considered to be skew lines if they are not ______________ and do not
__________________.
Similarly, two planes that do not intersect are ___________________________________.
Small directed triangles are used to show that lines are parallel. The symbol || means “is
parallel to” as in m || n .
Example 1: Think of each segment in the figure as part of a line.
Which line(s) or plane(s) in the figure appear to fit the
description?
a. Plane parallel to plane EFG and containing point A.
b. Name a line through point E that appears skew to CD .
c. Lines perpendicular to CD containing point A.
d. Lines parallel to CD containing point A.
Postulate 13 (Parallel Postulate): If there is a line and a point not on a line, then there is
exactly one line through the point parallel to the given line.
Example:
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Postulate 14 (Perpendicular Postulate): If there is a line and a point not on the line, then
there is exactly one line through the point perpendicular to the given line.
Example:
Example 2: The given line markings show how the roads are related to one another.
a. Name a pair of perpendicular lines.
b. Name a pair of parallel lines.
c. Is FE parallel to MN ? Use the parallel postulate to
explain why or why not.
Practice: Use the markings in the diagram. Name a pair of
perpendicular lines and a pair of parallel lines.
Angles and Transversals
A transversal is a line that intersects two or more coplanar lines at different points.
Example:
Two angles are _____________________ angles if they have corresponding positions.
Example:
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Two angles are ______________________________ angles if they lie between the two lines on
opposite sides of the transversal.
Example:
Two angles are _____________________________ angles if they lie outside the two lines and on
opposite sides of the transversal.
Example:
Two angles are ____________________________ angles if they lie between the two lines and on
the same side of the transversal.
Example:
Example 3: Identify all pairs of angles of the given type:
a. Corresponding
c. Alternate exterior
b. Alternate interior
d. Consecutive interior
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Practice: Classify the pair of numbered angles in each diagram:
3.2 Use Parallel Lines and Transversals
Postulate 15 (Corresponding Angles Postulate): If two parallel lines are cut by a transversal,
then the pairs of corresponding angles are congruent.
Example:
Example 1: Find the measure of the
angle of ( e) = 73o.
a and explain how the angle is related to the given
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Theorem 3.1 Alternate Interior Angles Theorem: If two parallel lines are cut by a
transversal, then the pairs of alternate interior angles are congruent.
Example:
Theorem 3.2 Alternate Exterior Angles Theorem: If two parallel lines are cut by a
transversal, then the pairs of alternate exterior angles are congruent.
Example:
Theorem 3.3 Consecutive Interior Angles Theorem: If two parallel lines are cut by a
transversal, then the pairs of consecutive interior angles are supplementary.
Example:
Example 2: Find the value of x.
Lines a and b are parallel.
m 1 = 116°
m 2 = (x + 5)°
Practice: Use the diagram to answer the following
questions:
a. Lines m and n are parallel. If m 1 = 109 , find m 4.
b. Lines m and n are parallel. If m 3 = 76 , and m 8 =
(2x + 8) , what is the value of x?
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Example 3: Prove that if two parallel lines are cut by a transversal, then the pairs of
alternate interior angles are congruent.
Example 4: Given that L1 and L2 are parallel, which of the following angles has the same
measure as d?
3.3 Prove Lines are Parallel
Postulate 16 (Corresponding Angles Converse): If two lines are cut by a transversal so the
corresponding angles are congruent, then the lines are parallel.
Example:
Example 1: Find the value of x that makes m n.
Example 2: Find the value of x that makes m n.
Example 3: Is there enough information in the
diagram to conclude that m n?
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Theorem 3.4 Alternate Interior Angles Converse: If two lines are cut by a transversal so the
alternate interior angles are congruent, then the lines are parallel.
Example:
Theorem 3.5 Alternate Exterior Angles Converse: If two lines are cut by a transversal so the
alternate exterior angles are congruent, then the lines are parallel.
Example:
Theorem 3.6 Consecutive Interior Angles Converse: If two lines are cut by a transversal so
the consecutive interior angles are supplementary, then the lines are parallel.
Example:
Example 4: Prove that if two lines are cut by a transversal so the alternate interior angles
are congruent, then the lines are parallel.
Theorem 3.7 Transitive Property of Parallel Lines: If two lines are parallel to the same line,
then they are parallel to each other.
Example:
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Example 5: Use the diagram below in the following exercise.
How would you show that the lines f and g are parallel?
Example 6: Each step in a staircase is parallel to the step immediately above it. The bottom
step is parallel to the ground. Explain why the top step is parallel to the ground.
3.4 Find and Use Slopes of Lines
The _____________ of a nonvertical line is the ratio of vertical change (rise) to horizontal
change (run) between any two points on the line. Slope is denoted by m and the formula is:
m = ________ = ________________ = ________ .
Slope of Lines in the Coordinate Plane:
Negative slope: Falls from left to right (down and to the right).
Positive slope: Rises from left to right (up and to the right).
Zero slope: Horizontal
Undefined slope: Vertical
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Example 1: Find the slope of the line that passes through the points (−4, 7) and (3, −7).
Example 2: Find the slope of line a.
You can compare slopes of lines to determine if they are ____________________ or
____________________.
Postulate 17 (Slopes of Parallel Lines): In a coordinate plane, two non-vertical lines are
parallel if and only if they have the same slope.
Postulate 18 (Slopes of Perpendicular Lines): In a coordinate plane, two non-vertical lines
are perpendicular if and only if the product of their slopes is -1.
Example 2: Find the slope of each line. Which lines are parallel?
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Example 3: Line h passes through the points (3,0) and (7,6). Graph the line perpendicular to h
that passes through the point (2,5).
Example 4: Line m passes through (−5, 5) and (−6, 1). Line t passes through (5, 6) and (4, 2). Are
the two lines parallel?
Example 5: During the climb of a roller coaster, you move 45 feet upward for every 78 feet
you move horizontally. At the crest of the hill, you have moved 390 feet forward. How high
is the roller coaster at the top of the climb?
Example 6: Determine if the lines are parallel or perpendicular.
Line a: y = 5x − 4
Line b: −40x + 8y = 2
Example 7: Find the slope of the line that passes through the points.
(1, 5), (5, 8)
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3.5 Write and Graph Equations of Lines
Linear equations may be written in different forms. The general form of a linear equation in
slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
Example 1: Write the equation which represents the line shown in slope-intercept form.
Write all numbers in your answer as simplified fractions or integers.
Example 2: Write an equation of the line that passes through the point (−3, −3) and is parallel
to the line y = 5x − 6.
Example 3: Write an equation of the line that passes through the point (9, −2) and is
perpendicular to the line y = 9x + 3.
Example 4: The graph models the total cost of joining a gym. Write an equation of the line.
What was the initial cost to join the gym?
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Example 5: Given the equation x + 5y =10 determine the following:
The x-intercept is:
The y-intercept is:
Example 6: You can rent DVDs at a local store for $6.00 each. An Internet company offers
a flat fee of $13.50 per month for as many rentals as you want. How many DVDs do you need
to rent to make the online rental a better buy?
Example 7: Write an equation of the line that passes through the point (2,-5) and is parallel
to the line y 
1
x4.
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Example 8: Write an equation of the line that passes through the point (0, −1) and is parallel
to the line y = 5x + 5.
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3.6 Prove Theorems about Perpendicular Lines
Theorem 3.8: If two lines intersect to form a linear pair of congruent angles, then the lines
are perpendicular.
Example:
Theorem 3.9: If two lines are perpendicular, then they intersect to form four right angles.
Example:
Example 1: In the diagram below, BD is perpendicular to AB. What can you conclude about
angle 4 and angle 5?
Theorem 3.10: If two sides of two adjacent acute angles are perpendicular, then the angles
are complementary.
Example:
Example 2: Prove that if two sides of two adjacent acute angles are perpendicular, then the
angles are complementary.
Example 3: Given that ABC
conclude about 3 and 4?
ABD, what can you
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Theorem 3.11 Perpendicular Transversal Theorem: If a transversal is perpendicular to one or
two parallel lines, then it is perpendicular to the other.
Example:
Theorem 3.12 Lines Perpendicular to a Transversal Theorem: In a plane, if two lines are
perpendicular to the same line, then they are parallel to each other.
Example:
Example 4: Determine which lines, if any, must be parallel in the
diagram. Explain your reasoning.
Example 5: Is b c?
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