Download Chapter 3 Angles and Lines

3.1 Angles and Their Measures
Chapter 3
Angles and Lines
n
Angle
n
Sides
n The union
of two rays that have the same endpoint
n (of an angle) the rays
n
that form it
Vertex
n (of an angle) the common endpoint of the two rays.
3.1 Angles and Their Measures
Names of angles
<A or <BAC or <CAB Not <ABC (why not?)
n
3.1 Angles and Their Measures
n
n
n
B
Can you call this <B?
What is another name for <1?
What is anther name for <2?
A
C
D
C
A
1
2
B
1
3.1 Angles and Their Measures
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Measure of Angle
3.1 Angles and Their Measures
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m<ABC = measured in degrees.
n Full circle is 360º
n Straight angle is 180 º
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Angle Addition Property
uuu
r
n If VC (except for point V) is in the interior of
<AVB, then
m<AVC = m<CVB = m<AVB
C
n
B
Tool used to measure degrees is half of a circle
commonly called a _______________.
A
V
3.1 Angles and Their Measures
n
n
n
m<DEC?
m<CEA?
m<AED?
3.1 Angles and Their Measures
n
n
B
A
85º
E
D
20 º
Bisector
C
n
A point, line, ray or plane which divides a segment, angle, or
figure into two equal parts.
uur
uur
VR is the bisector of <PVQ if and only if VR (except
for point V) is in the interior of <PVQ and m<PVR =
m<RVQ
n
n
What would the above figure look like?
Say m<PVR = 3x + 2 and m<QVR = 4x –8 solve for x.
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3.1 Angles and Their Measures
n
Assumptions from drawings page 128
n
1.
2.
3.
3.1 Angles and Their Measures
n
From a figure you can assume:
assume:
Collinearity and betweenness of points drawn on
lines.
Intersections of lines at a given point.
Points in the interior of an angel, on an angel, or in
the exterior of an angle.
From a figure, you cannot assume:
1.
2.
3.
4.
3.3 Properties of Angles
n
Types of Angles (m is the measure)
Zero angle m = 0º
n Acute angle 0º < m < 90º
n Right angle m = 90º
n Obtuse angle 90º < m < 180º
n Straight angle m = 180º
3.3 Properties of Angles
n
n
n
Take out the Protractor worksheet Activity 3.1
and label each angle with one of the above
terms.
Collinearity of points that are not drawn on lines
Parallel Lines
Exact measures of angles and lengths of segments.
Measures of angles or lengths of segments are
equal
Complementary Angle
n
Two angles that add to make 90º
n Say m<1
n
= 20º what is the measure of its complement?
Supplementary Angle
n
Two angles that add to 180º
n Say m
<1 = 20º what is the measure of its supplement?
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3.3 Properties of Angles
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3.3 Properties of Angles
Adjacent angles
n
n
n
n
Two nonnon-straight and
nonnon-zero angles are
adjacent angles if and only
if a common side is
interior to the angle
formed by the nonnoncommon sides
What is the name of the
common side?
What is the name of the
adjacent angles?
Linear pair
n
B
C
n
A
n
n
V
n
Two adjacent angles form a
linear pair if and only if
their nonnon-common sides are
opposite rays.
<1 and <2 are a linear pair.
Thm.
Thm. <1 + <2 =180º What
is another name for this
condition?
m<1 = 50º, m<2 = ?
m<1 = y, m<2 = ?
3.3 Properties of Angles
n
2
3.3 Properties of Angles
Vertical angles
n
1
m<1 + m<2 = ?
n m<1 + m<4 = ?
n So
m<1 +m<2 = m<3 + m<4
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Two nonnon-straight angles are vertical angles if and
only if the union of their sides is 2 lines. (formed by
intersecting lines)
m<2 = m<4
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1
n
3
2
4
1
Thm.
Thm.
n
3
2
If two angles are vertical angles then
they have equal measure.
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3.4 Algebra Properties Used in
Geometry
3( x + 7) - 5 x = - 6( x - 5)
n
n
Solve the given equation
Identify the following steps:
Distributive Property
n Combine like terms
n Addition property of Equality
n Multiplication property of Equality
3.4 Algebra Properties Used in
Geometry
Draw AD
n
Given
AB = CD
n
Addition Property of
Equality
AB + BC = BC + CD
n
Segment Addition
Property
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3.4 Algebra Properties Used in
Geometry
n
n
What happens when you solve -2x>6?
Postulates of Equality and Inequality
n
If a and b are positive and a + b = c ,
then c > a and c > b.
n Ex.
n
AB + BC = 10, then AB <10 AND BC < 10
Substitution Property
n
If a = b, then a may be substituted for b in any
expression.
n Ex.
90 –x, x=30
AC = BD
3.5 OneOne-Step Proof Arguments
n
n
OneOne-Step Proof Arguments –a given and one
conclusion.
Proof Argument
n
A conditional is a sequence of justified conclusions.
Starting with the antecedent and ending with the
consequent.
n Ex.
0.
1.
2.
3y -7 = 8 Given
3y = 15 Addition Property of Equality
y = 5 Multiplication Property of Equality
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3.5 OneOne-Step Proof Arguments
n
Three Types of Justification in a Proof
Argument.
1.
2.
3.
3.5 OneOne-Step Proof Arguments
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Definition of Obtuse Angles
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Postulates –assumed properties
Definitions –Defining properties
Theorems –Deduced (previously proven)
properties.
Or
Example
m<ABC = 120º Justify that <ABC is obtuse.
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3.6 Parallel Lines
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n
3-6 Corresponding Angle Activity
Need Patty paper, straight edge, worksheet
An angle whose measure is greater than 90 and less
than 180.
If an angle measure greater than 90 and less than 180
it is obtuse.
3.6 Parallel Lines
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Corresponding Angles
Postulate –
suppose two
coplanar lines are cut by
a transversal.
n
n
If two corresponding
angles have the same
measure, then the lines
are parallel
If the lines are parallel,
then corresponding angles
have the same measure.
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Note abbreviations:
l
1
2
4
3
5
m
6
8
7
6
3.6 Parallel Lines
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Def. Slope of the LineLinen
3.6 Parallel Lines
n
the slope of the line through
( x1 , y1 ) and ( x2 , y2 ), with x1 ¹ x2 ,
Parallel lines and Slopes TheoremTheoremn
n
y - y
is 2 1
x2 - x1
Transitivity of Parallelism TheoremTheoremn
3.7 Perpendicular Lines
n
n
1.
2.
3.
WarmWarm-up
Consider points A = (5, -4), B = (8, 2), and
C = (2, 5)
suu
r
suu
r
Find the slopes of AB and BC
Find the product of the slopes.
Graph the lines and find m<CBA.
Two nonvertical lines are parallel if and only if they
have the same slope.
In a plane, if line l, is parallel to line m, and line m is
parallel to line n, then line l is parallel to line n.
3.7 Perpendicular Lines
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Perpendicular Lines –
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Two segments, rays, or
lines are perpendicular if
and only if the lines
containing them form a
90º angle.
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Do we need to label
more than one angle?
m
n
m^ n
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3.7 Perpendicular Lines
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3.7 Perpendicular Lines
Two Perpendicular’
s
TheoremTheoremn
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Perpendicular to Parallels TheoremTheoremn
If two coplanar lines l and
m are each perpendicular
to the same line, then they
are ________ to each
other.
m
n
n
In a plane, if a line is perpendicular to one fo the two
parallel lines, then it is also perpendicular to the
other.
Perpendicular Lines and Slopes TheoremTheoremn
Two nonvertical lines are perpendicular if and only if
the product of their slopes is -1.
l
3.8 Drawing Parallel and
Perpendicular Lines
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n
MUST bring in compass and straight edge
Bisector of a segmentsegmentn
is its midpoint, or any line, ray or segment which
intersects the segment only at its midpoint. Only
one perpendicular bisector.
3.8 Drawing Parallel and
Perpendicular Lines
n
n
m
n
n
Constructions –a precise way of drawing which
uses specific tools and follows specific rules.
Two Tools allowed
1. Compass
2. Unmarked straightstraight-edge
(Constructions may also use Patty paper when
allowed.)
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3.8 Drawing Parallel and
Perpendicular Lines
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n
n
n
n
n
n
Constructions
1. Segment copy
2. Perpendicular bisector of a line segment
3. Patty paper of #2
4. Angle bisector (traditional/patty paper)
5. Angle Copy (traditional/patty paper)
6. Construct a line parallel to another line
through a given point.
3.8 Drawing Parallel and
Perpendicular Lines
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1.
2.
3.
4.
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Construct a line parallel to another line
through a given point. (traditional only)
draw any line m
draw arcs –same setting using vertex Q, and P
copy angles
draw L
Why is it parallel?
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