Download 8.6 Relationships Between Angle Measures

Geometry
CHAPTER
8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
Basic Geometric Figures
Perimeter
Area
Circles
Volume and Surface Area
Relationships Between Angle Measures
Congruent Triangles and Properties of
Parallelograms
8.8 Similar Triangles
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 2
8.6
Relationships Between Angle
Measures
OBJECTIVES
a Identify complementary and supplementary angles
and find the measure of a complement or a
supplement of a given angle.
b Determine whether segments are congruent and
whether angles are congruent.
c Use the Vertical Angle Property to find measures
of angles.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 3
8.6
Relationships Between Angle
Measures
OBJECTIVES
d Identify pairs of corresponding angles, interior angles,
and alternate interior angles and apply properties of
transversals and parallel lines to find measures of
angles.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 4
8.6
Relationships Between Angle
Measures
COMPLEMENTARY ANGLES
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Slide 5
8.6
Relationships Between Angle
Measures
Identify complementary and supplementary angles
a and find the measure of a complement or a
supplement of a given angle.
If two angles are complementary, each is an acute angle.
When complementary angles are adjacent to each other,
they form a right angle.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 6
8.6
a
Relationships Between Angle
Measures
Identify complementary and supplementary angles and find the
measure of a complement or a supplement of a given angle.
EXAMPLE 2
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 7
8.6
Relationships Between Angle
Measures
SUPPLEMENTARY ANGLES
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Slide 8
8.6
a
Relationships Between Angle
Measures
Identify complementary and supplementary angles and find the
measure of a complement or a supplement of a given angle.
EXAMPLE 4
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 9
8.6
Relationships Between Angle
Measures
Determine whether segments are congruent and
b
whether angles are congruent.
Congruent figures have the same size and shape. They
fit together exactly.
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Slide 10
8.6
Relationships Between Angle
Measures
CONGRUENT SEGMENTS
Two segments are congruent if and only if they have
the same length.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 11
8.6
Relationships Between Angle
Measures
Determine whether segments are congruent and
b
whether angles are congruent.
EXAMPLE 6
Which pairs of segments are congruent? Use a ruler.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 12
8.6
Relationships Between Angle
Measures
CONGRUENT ANGLES
Two angles are congruent if and only if they have the
same measure.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 13
8.6
Relationships Between Angle
Measures
Determine whether segments are congruent and
b
whether angles are congruent.
EXAMPLE 8
Which pairs of angles are congruent? Use a protractor.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 14
8.6
Relationships Between Angle
Measures
Determine whether segments are congruent and
b
whether angles are congruent.
If two angles are congruent, then their supplements are
congruent and their complements are congruent.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 15
8.6
Relationships Between Angle
Measures
VERTICAL ANGLES
Two nonstraight angles are vertical angles if and only
if their sides form two pairs of opposite rays.
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Slide 16
8.6
Relationships Between Angle
Measures
THE VERTICAL ANGLE PROPERTY
Vertical angles are congruent.
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Slide 17
8.6
Relationships Between Angle
Measures
Use the Vertical Angle Property to find measures
c
of angles.
EXAMPLE 9
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Slide 18
8.6
Relationships Between Angle
Measures
Use the Vertical Angle Property to find measures
c
of angles.
EXAMPLE 9
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Slide 19
8.6
Relationships Between Angle
Measures
TRANSVERSAL
A transversal is a line that intersects two or more
coplanar lines in different points.
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Slide 20
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
Corresponding Angles
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 21
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
Interior Angles
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 22
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
Alternate Interior Angles
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 23
8.6
d
Relationships Between Angle
Measures
Identify pairs of corresponding angles, interior angles, and
alternate interior angles and apply properties of transversals
and parallel lines to find measures of angles.
EXAMPLE 10
Identify all pairs of corresponding angles, all interior
angles, and all pairs of alternate interior angles.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 24
8.6
d
Relationships Between Angle
Measures
Identify pairs of corresponding angles, interior angles, and
alternate interior angles and apply properties of transversals
and parallel lines to find measures of angles.
EXAMPLE 10
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 25
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
Properties of Parallel Lines
1. If a transversal intersects two parallel lines, then the
corresponding angles are congruent.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 26
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
2. If a transversal intersects two parallel lines, then the
alternate interior angles are congruent.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 27
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
3. In a plane, if two lines are parallel to a third line, then
the two lines are parallel to each other.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 28
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
4. If a transversal intersects two parallel lines, then the
interior angles on the same side of the transversal are
supplementary.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 29
Relationships Between Angle
Identify pairs
of corresponding angles, interior angles,
Measures
8.6
and alternate interior angles and apply properties of
d
transversals and parallel lines to find measures of
angles.
5. If a transversal is
perpendicular to one of
two parallel lines, then
it is perpendicular to
the other.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 30
8.6
d
Relationships Between Angle
Measures
Identify pairs of corresponding angles, interior angles, and
alternate interior angles and apply properties of transversals
and parallel lines to find measures of angles.
EXAMPLE 11
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 31
8.6
d
Relationships Between Angle
Measures
Identify pairs of corresponding angles, interior angles, and
alternate interior angles and apply properties of transversals
and parallel lines to find measures of angles.
EXAMPLE 11
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 32
8.6
d
Relationships Between Angle
Measures
Identify pairs of corresponding angles, interior angles, and
alternate interior angles and apply properties of transversals
and parallel lines to find measures of angles.
EXAMPLE 12
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 33