3.1 Lines and Angles
Parallel lines: two coplanar lines which do not intersect.
Skew lines: two lines which are not coplanar and do not intersect.
Parallel planes: two planes which do not intersect.
Parallel Postulate: if there is a line and a point not on
the line, then there is exactly one line through the
point parallel to the given line.
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.
Transversal: a line that intersects two or more coplanar lines at
different points.
Corresponding angles: two angles if they occupy corresponding
positions.
Alternate Exterior angles: two angles that lie outside the two lines on
opposite sides of the transversal.
Alternate interior angles: two angles that lie inside the two lines on
opposite sides of the transversal.
Consecutive interior angles: two angles that lie between the two lines
on the same side of the transversal.
1 2
3 4
5
6
7
8
Draw two lines and a transversal.
Label the angles a, b, c, d, e, f, g and h.
Which angles are corresponding angles?
Which angles are alternate interior angles?
Which angles are alternate exterior angles?
Which angles are same side interior angles?
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3.2 Proof and Perpendicular Lines
Types of Proof
Two column proof: lists numbers and statements in two column format.
Flow proof: uses arrows to show the logical flow of the statements.
Paragraph proof: conversational form of proof in a paragraph form.
If two lines intersect to form a linear pair of
congruent angles, then the lines are perpendicular.
If two sides of two adjacent acute angles are
perpendicular, then the angles are complementary.
If two lines are perpendicular, then they intersect
to form four right angles.
3.3 Parallel Lines and Transversals
If two parallel lines are cut by a transversal, then:
the corresponding angles are congruent.
the alternate interior angles are congruent.
the alternate exterior angles are congruent.
the consecutive interior angles are supplementary.
Given m<5 = 52 and mlln find the measure of each of the other angles.
Find the value of x.
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3.4 Proving Lines are Parallel
**Converses are when the if and then statements are reversed.**
Corresponding Angles Converse: if the corresponding angles are
congruent, then the lines are parallel.
Alternate Interior Angles Converse: if the alternate interior angles
are congruent, then the lines are parallel.
Alternate Exterior Angles Converse: if the alternate exterior angles
are congruent, then the lines are parallel.
Consecutive Interior Angles Converse: if the consecutive interior
angles are supplementary, then the lines are parallel.
Identifying Parallel Lines
Which rays are parallel?
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Are the rays parallel?
3.5 Using Properties of Parallel Lines
Proving Two Lines are Parallel
Given: m ll n and n ll k
Prove m ll k
Statements
Reasons
1. m ll n
1. Given
2. <1 = <2
2. Corresponding
3. n ll k
3. Given
4. <2 = <3
4. Corresponding
5. <1 = <3
5. Transitive
6. m ll k
6. Corresponding
1
m
2
n
3
k
Angles Postulate
Angles Postulate
Angles Converse
Theorem 3.11: If two lines are parallel to the same line,
then they are parallel to each other.
If p ll q and q ll r, then p ll r
Theorem 3.12: In a plane, if two lines are perpendicular
to the same line, then they are parallel to each other.
If m p and n p, then m ll n
Name the ways you can prove two lines are parallel.
1.
2.
3.
4.
5.
6.
These will be the reasons (postulates and theorems) in your homework.
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3.6 Parallel Lines in the Coordinate Plane
ALGEBRA REVIEW!!!
Slope = rise / run
y2  y1
slope = m = x  x
2
1
Find the slope of the line that passes through the given points.
(0,6) and (5,2)
(3,0) and (2,-3)
(-5,-2) and (-2,-7)
Find the slope of the lines.
3
5
-6
5
3
2
Postulate 17: In a coordinate plane, two nonvertical lines are parallel if
and only if they have the same slope. Any two vertical lines are
parallel.
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3.6 Continued: Writing Equations of Parallel Lines
In Algebra you learned the slope-intercept form: y = mx + b
where m is the slope and b is the y-intercept.
Write an equation in slope-intercept form.
Slope = 3
slope = 2/3
slope = -1/2
y-int = 5
y-int = -3
y-int = 3/5
through (2,3) with m = 5
ll to y = 3x - 1
through (2,5)
through (-2,4) with m = -3
ll to y = -2x + 5
through (1,-2)
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ll to y = (1/3)x – 8
through (3,2)
3.7 Perpendicular Lines in the Coordinate Plane
Postulate 18: In a coordinate plane, two nonvertical lines are
perpendicular if and only if the product of their slopes is -1.
Vertical and Horizontal lines are perpendicular.
**Their slopes are opposite reciprocals.**
If a line has a slope of m, the perpendicular line has slope of
1
m
What is the slope of the line perpendicular to the given slope:
2/3
3/5
-4/5
6
-3
Decide whether the lines are perpendicular.
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3.7 Continued…
Decide whether the lines are perpendicular.
y=¾x+5
y = (- 2/3) x – 6
y = (- 4/3) x +2
y = (- 3/2) x -1
y=-2x
y = ½ x -9
Write the equation of the line perpendicular to the given line and
through the given point.
Y = ½ x +5, P(4,1)
y = (-2/3)x -3, P(6,-3)
Y= 3x – 5, P(3,-4)
y = -x + 4, P(2,2)
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