Chapter 9
Lesson 9-1
Introduction to Geometry:
Points, Lines, and Planes
Sample
∙
A
A
Symbol
Point
Point A
A location in space. It has no size.
Line
AB
BA
or n
A series of points that extends in
opposite directions without end. A
lowercase letter can name a line.
B
Plane
M
D
Name
C
P
Q
Description
A flat surface with no thickness. It
ABCD or contains many lines and extends
M
without end in the directions of all its
lines.
Line
Segment
or
segment
PQ
QP
A part of a line. It has two endpoints.
PQ represents the length of PQ.
Ray
CR
A part of a line. It has exactly one
endpoint. Name its endpoint first.
R
C
Intersecting, Parallel, and Skew Lines
• Two lines that lie in the
same plane and do not
intersect are parallel.
Use the symbol ║to
indicate “is parallel to”.
• Two lines intersect if
they have exactly one
point in common.
• Skew lines are lines
that do not lie in the
same plane.
AB ║ PQ
EF intersects BF
AB and DE are
skew
Parts of an Angle
• An angle is formed by two rays with a
common endpoint.
• The rays are the sides of the angle.
• The common endpoint is the vertex.
A
Ray
●
Angle
B
Endpoint
or Vertex
●
C
Classifying Angles
• An acute angle is
less than 90○.
Acute Angle
• A right angle is 90○.
Right Angle
• An obtuse angle is
○
greater than 90 and
○
less than 180 .
Obtuse Angle
• A straight angle is
equal to 180○.
Straight Angle
Lesson 9-2
Angle Relationships
and Parallel Lines
Adjacent and Vertical Angles
• Adjacent angles
share a vertex and a
side but no points in
their interiors.
• Vertical angles are
formed by two
intersecting lines
and are opposite
each other.
1
Common Side
3
4
2
Angles 1 & 2 are vertical angles.
Angles 3 & 4 are vertical angles.
Angle Relationships
• If the sum of the measures
○
of two angles is 90 , the
angles are complementary.
Complementary Angles
• If the sum of the measures
○
of two angles is 180 , the
angles are supplementary.
Supplementary Angles
Relating Angles and Parallel
Lines
A line that
intersects
two other
lines in
different
points is a
transversal.
When a transversal intersects
two parellel lines,
corresponding and alternate
interior angles are congruent.
Alternate interior angles are in the
interior of a pair of lines and on
opposite sides of the transversal.
d and e are alternate interior
angles.
Corresponding angles lie on the
same side of the transversal and in
corresponding positions. d and
h are corresponding angles.
Lesson 9-3
Classifying Polygons
Classifying Triangles
• A triangle is a polygon with three sides.
Acute triangle
three acute sides
Right triangle
one right angle
Equilateral triangle
three congruent sides
Isosceles triangle at least
two congruent sides
Obtuse triangle
one obtuse angle
Scalene triangle
no congruent sides
Classifying Quadrilaterals
Quadrilateral
four sides
Parallelogram
both pairs of
opposite sides
parallel
Rectangle
○
four 90
angles
Trapezoid
exactly one pair of
parallel sides
Rhombus
four congruent
sides
Square
○
four 90 angles and
four congruent sides
Classifying Quadrilaterals Cont.
• All parallelograms have opposite sides
parallel.
• Parallelograms include rectangles,
rhombuses, and squares.
• Quadrilaterals that have four right
angles include the rectangles and
squares.
Regular Polygons
• A regular polygon has all sides
congruent and all angles congruent.
• The formula for the perimeter of a
regular polygon is P = number of sides
 the length of the sides.
Triangle
Pentagon
Square
Hexagon
Quadrilaterals and Their
Properties
Quadrilateral Quest: Do You Know Their
Properties?
Lesson 9-5
Congruence
Congruent Triangles
• Congruent figures have the same size
and shape, and their corresponding
parts have equal measures.
• Triangles are congruent when all
corresponding sides and interior
angles are congruent.
• You use corresponding parts of
triangles to identify congruent
triangles.
Congruent Triangles
Side-Side-Side (SSS)
Angle-Side-Angle (ASA)
If three sides of one triangle are congruent
to three sides of a second triangle, the two
triangles are congruent.
If two angles and the included side of one
triangle are congruent to two angles and
the included side of another triangle, the
triangles are congruent.
Side-Angle-Side (SAS)
Congruent figures have
the same size and
shape, and their
corresponding parts have
equal measures.
If two sides and the included angle are
congruent to two sides and the included
angle of a second triangle, the two
triangles are congruent.
Lesson 9-6
Circles
Circle
Radius is a
segment that has
one endpoint at
the center and the
other point on the
circle.
Diameter is a
chord that passes
through the
center of a circle.
Circumference is
the distance
around the circle.
Chord is a
segment whose
endpoints are on
the circle.
Circumference of a Circle
• The circumference of a circle is π
times the diameter.
C=πd
C=2πr
C=πd
Write the formula
C ≈ (3.14)6
Replace π with 3.14
and d with 6
= 18.84
Simplify
6 ft
Making a Circle Graph
• To make a circle graph, you find the measure of
each central angle.
• A central angle is an angle whose vertex is the
center of a circle.
○
• There are 360 in a circle.
• Use proportions to find the measures of the central
angles.
• Use a compass to draw a circle.
20 = _r
100
360
r = 72
○
25 = _r_
100
r = 90
360
○
•Draw the central angles with a
protractor.
•Label each section.
•Add a title and necessary
information.
Lesson 9-7
Constructions
Construction Vocabulary
Perpendicular lines, segments, or
rays intersect to form right angles.
A perpendicular bisector is a line,
segment, or ray that is perpendicular
to the segment it bisects.
A segment bisector is a line,
segment, or ray that divides a
segment into two congruent
segments.
Construction Vocabulary Cont.
An angle bisector is
a ray that divides an
angle into two
congruent angles.
Steps for Constructing a(n) . . .
Congruent Segment
Pearson Prentice Hall Mathematics Video
Congruent Angle
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Perpendicular Bisector
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Angle Bisector
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Lesson 9-8
Translations
Translation Vocabulary
• You perform a translation by sliding,
flipping, or turning an object.
• A transformation is a change of position or
size of a figure.
• A translation is a transformation that moves
points the same distance and in the same
direction.
• The figure you get after a transformation is
called the image.
1
• Use prime notation (A ) to name the image
of a point.
Examples of Translations
Example of a Slide
Example of a Flip
Translating a Figure
Example of a Turn
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Writing a Rule to
Describe a Translation
Pearson Prentice Hall Mathematics Video
Lesson 9-9
Symmetry and Reflections
Symmetry
• A figure has reflectional symmetry when
one half is a mirror image of the other half.
• A line of symmetry divides a figure with
reflectional symmetry into two congruent
halves.
Reflections
• A reflection is a
transformation that
flips a figure over a
line of reflection.
• The reflected figure,
or image, is congruent
to the original figure.
• Together, an image
and its reflection have
line symmetry, the line
of reflection being the
line of symmetry.
Graphing Reflections of a Shape
Pearson Prentice Hall Mathematics Video
Lesson 9-10
Rotations
Rotations
• A rotation is a
transformation that
turns a figure about a
fixed point called the
center of rotation.
• The angle measure of
the rotation is the
angle of rotation.
Rotational Symmetry
• A figure has rotational
○ symmetry if
you can rotate it 360 , or less, so that
its image matches the original figure.
• The angle (or its measure)
through which the figure
rotates is the angle of
rotation.