Introduction to Geometry: Points, Lines, and Planes

1
Introduction to Geometry Points, Lines, and
Planes
PRE-ALGEBRA LESSON 9-1
The telephone company is installing telephone
lines for ten buildings. Each building is to be
connected to each of the other buildings with one
line. How many telephone lines are needed?
45
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Introduction to Geometry Points, Lines, and
Planes
PRE-ALGEBRA LESSON 9-1
(For help, go to Lesson 2-8.)
Describe the number-line graph of each
inequality. 1. a 3 2. a 0 3. a
5 4. a 2
Check Skills Youll Need
9-1
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Introduction to Geometry Points, Lines, and
Planes
PRE-ALGEBRA LESSON 9-1
Solutions 1. The graph is a line that starts
at 3 and extends to the right without end.
2. The graph is a line that starts at 0 and
extends to the left without end. 3. The graph
is a line that starts at 5 and extends to the
left without end. 4. The graph is a line that
starts at 2 and extends to the right without
end.
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Introduction to Geometry Points, Lines, and
Planes
PRE-ALGEBRA LESSON 9-1
Quick Check
a. four points
b. four different segments
c. five other names for KI
d. five different rays
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Introduction to Geometry Points, Lines, and
Planes
PRE-ALGEBRA LESSON 9-1
You are looking directly down into a wooden
crate. Name each of the following.
a. four segments that intersect PT
b. three segments parallel to PT
c. four segments skew to PT
Quick Check
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Introduction to Geometry Points, Lines, and
Planes
PRE-ALGEBRA LESSON 9-1
Draw two intersecting lines. Then draw a segment
that is parallel to one of the intersecting lines.
Use the lines on notebook or graph paper.
Quick Check
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Introduction to Geometry Points, Lines, and
Planes
PRE-ALGEBRA LESSON 9-1
Use the figure. Name each of the
following. 1. four points 2. another name
for EA 3. three different rays 4. three
segments that are parallel to HG 5. four
segments that are skew to CG 6. four segments
that intersect AE
A, B, C, D
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Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
The Jackson County Bird Sanctuary has three times
as many owls as hawks. It has 40 hawks and owls
in all. How many of each are in the sanctuary?
30 owls, 10 hawks
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Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
(For help, go to Lesson 7-5.)
Solve. 1. n 45 180 2. 75 x 90 3. 3y
2y 90 4. 2a 15 a 45
Check Skills Youll Need
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Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
Solutions 1. n 45 180 2.
75 x 90 n 45 45 180
45 75 75 x 90 75 n
135 x 15 3. 3y 2y
90 4. 2a 15 a 45 3y 2y 2y
2y 90 2a a 15 a a 45 y
90 a 15 45 a 15 15 45
15 a 30
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Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
Find the measure of 3 if m 4 110.
Quick Check
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Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
In the diagram, p q. Identify each of the
following.
a. congruent corresponding angles
b. congruent alternate interior angles
Quick Check
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Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
145
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Classifying Polygons
PRE-ALGEBRA LESSON 9-3
Draw an example of each kind of angle and
describe its properties. a. acute angle A b.
right angle R c. obtuse angle O
Check students drawings.
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Classifying Polygons
PRE-ALGEBRA LESSON 9-3
(For help, go to Lesson 9-2.)
For the angle measures given, classify the angle
as acute, right, or obtuse. 1. 85 2. 95 3. 16
0 4. 90 5. 36 6. 127
Check Skills Youll Need
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Classifying Polygons
PRE-ALGEBRA LESSON 9-3
Solutions 1. acute 2. obtuse 3. obtuse 4. ri
ght 5. acute 6. obtuse
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Classifying Polygons
PRE-ALGEBRA LESSON 9-3
Classify the triangle by its sides and angles.
The triangle has no congruent sides and one
obtuse angle.
The triangle is a scalene obtuse triangle.
Quick Check
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Classifying Polygons
PRE-ALGEBRA LESSON 9-3
Name the types of quadrilaterals that have at
least one pair of parallel sides.
All parallelograms and trapezoids have at least
one pair of parallel sides.
Parallelograms include rectangles, rhombuses, and
squares.
Quick Check
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Classifying Polygons
PRE-ALGEBRA LESSON 9-3
A contractor is framing the wooden deck shown
below in the shape of a regular dodecagon (12
sides). Write a formula to find the perimeter of
the deck. Evaluate the formula for a side length
of 3 ft.
To write a formula, let x the length of each
side.
The perimeter of the regular dodecagon is   x
x x x x x x x x x x x.
Therefore a formula for the perimeter is P 12x.
P 12x Write the formula.
12(3)   Substitute 3 for x.
36 Simplify.
Quick Check
For a side length of 3 ft, the perimeter is 36 ft.
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Classifying Polygons
PRE-ALGEBRA LESSON 9-3
Name the following. 1. a type of triangle that
has at least two congruent sides and one right
angle 2. a type of quadrilateral that can have
opposite sides parallel and no right angles
3. Write a formula for the perimeter of a
regular heptagon (7 sides). Evaluate for a side
of 12 in.
isosceles right triangle
parallelogram, rhombus
P 7x 84 in.
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Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
Draw several different quadrilaterals. Connect
the midpoints of the sides of each figure. Write
a sentence explaining in what way the figures
inside the quadrilaterals are alike.
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Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
(For help, go to Lesson 9-3.)
Sketch each figure. 1. equilateral
triangle 2. rectangle 3. pentagon 4. hexagon 5.
octagon
Check Skills Youll Need
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Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
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Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
How many diagonals does a nonagon have?
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Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
(continued)
You can organize your results as you count the
diagonals. Do not count a diagonal twice. (The
diagonal from A to C is the same as the one from
C to A.) Then find the sum of the numbers of
diagonals.
Vertex
Number of Diagonals
6
A
6
B
5
C
4
D
E
3
F
2
G
1
H
0
I
0
Total
27
A nonagon has 27 diagonals.
Quick Check
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Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
Solve. 1. How many diagonals does a
quadrilateral have? 2. How many triangles can
you form if you draw all the diagonals from one
vertex of a pentagon? 3. How many triangles
can you form if you draw all the diagonals of a
rectangle?
2 diagonals
3 triangles
8 triangles
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Congruence
PRE-ALGEBRA LESSON 9-5
Replace the question marks with the correct
digits. a. 8 9 6.
15.96 b. 13. 0 . 4 2
4.122
9, 9, 7
6, 4, 9, 8
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Congruence
PRE-ALGEBRA LESSON 9-5
(For help, go to Lesson 6-3.)
Check Skills Youll Need
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Congruence
PRE-ALGEBRA LESSON 9-5
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Congruence
PRE-ALGEBRA LESSON 9-5
In the figure, TUV WUX.
a. Name the corresponding congruent angles.
b. Name the corresponding congruent sides.
Quick Check
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Congruence
PRE-ALGEBRA LESSON 9-5
List the congruent corresponding parts of each
pair of triangles. Write a congruence statement
for the triangles.
a.
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Congruence
PRE-ALGEBRA LESSON 9-5
(continued)
b.
Quick Check
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Congruence
PRE-ALGEBRA LESSON 9-5
SAS
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Circles
PRE-ALGEBRA LESSON 9-6
Solve the proportion
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Circles
PRE-ALGEBRA LESSON 9-6
(For help, go to Lesson 6-2.)
Solve each proportion. Round to the nearest whole
number where necessary. 1. 2.
3. 4.
Check Skills Youll Need
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Circles
PRE-ALGEBRA LESSON 9-6
Solutions 1. 36 2. 270 3. 54
4. 109
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Circles
PRE-ALGEBRA LESSON 9-6
Find the circumference of the circle.
The circumference of the circle is about 37.68 in.
Quick Check
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Circles
PRE-ALGEBRA LESSON 9-6
Make a circle graph for Jackies weekly budget.
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Circles
PRE-ALGEBRA LESSON 9-6
(continued)
Quick Check
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Circles
PRE-ALGEBRA LESSON 9-6
Draw a circle graph of the data.
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Circles
PRE-ALGEBRA LESSON 9-6
(continued)
Use a compass to draw a circle.
Quick Check
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Circles
PRE-ALGEBRA LESSON 9-6
Solve. 1. Find the circumference of a circle
with a diameter of 2.5 in. 2. Ten out of 22
students surveyed prefer milk with their
breakfast. Find the measure of the central
angle to represent this data in a circle
graph. 3. Draw a circle graph of the data.
about 7.85 in.
about 164
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Constructions
PRE-ALGEBRA LESSON 9-7
A rectangular field is three times as long as it
is wide. What are its width and length if the
perimeter is 600 yd?
width 75 yd length 225 yd
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Constructions
PRE-ALGEBRA LESSON 9-7
(For help, go to Lesson 9-1.)
Check Skills Youll Need
9-7
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Constructions
PRE-ALGEBRA LESSON 9-7
Solutions 1. point B 2. a line segment
with endpoints A and B 3. a ray with endpoint A
and containing point B 4. a line containing
points A and B
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Constructions
PRE-ALGEBRA LESSON 9-7
Construct a segment congruent to WX.
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Constructions
PRE-ALGEBRA LESSON 9-7
(continued)
Step 3 With the same compass setting, put
the compass tip on G. Draw an arc that
intersects the ray. Label the intersection
H.
GH WX
Quick Check
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Constructions
PRE-ALGEBRA LESSON 9-7
Construct an angle congruent to W.
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Constructions
PRE-ALGEBRA LESSON 9-7
(continued)
Quick Check
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Constructions
PRE-ALGEBRA LESSON 9-7
Construct the perpendicular bisector of WY.
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Constructions
PRE-ALGEBRA LESSON 9-7
(continued)
Step 2 Label the points of intersection S and
T. Draw ST. Label the intersection of ST and
WY point M.
Quick Check
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Constructions
PRE-ALGEBRA LESSON 9-7
Construct the bisector of W.
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Constructions
PRE-ALGEBRA LESSON 9-7
(continued)
Step 2  Put the compass tip at S. Draw an arc.
With the same compass setting, repeat with
the compass tip at T. Make sure the arcs
intersect. Label the intersection of the arcs
Z. Draw WZ.
Quick Check
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Constructions
PRE-ALGEBRA LESSON 9-7
13. Check students work.
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Translations
PRE-ALGEBRA LESSON 9-8
The short sides of a kite measure 54 cm each, and
the long sides each measure 78 cm. What is the
perimeter of the kite?
264 cm
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Translations
PRE-ALGEBRA LESSON 9-8
(For help, go to Lesson 1-10.)
Graph each point. 1. A(4, 3) 2. B(0,
2) 3. C(1, 4) 4. D(4, 2) 5. E(2, 3)
Check Skills Youll Need
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Translations
PRE-ALGEBRA LESSON 9-8
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Translations
PRE-ALGEBRA LESSON 9-8
Graph the image of BCD after a translation 3
units to the left and 4 units down.
Quick Check
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Translations
PRE-ALGEBRA LESSON 9-8
Use arrow notation to describe the
translation of X to X .
Quick Check
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Translations
PRE-ALGEBRA LESSON 9-8
Write a rule to describe the translation of
RST to R S T .
Horizontal translation 3 (2) 1
Vertical translation 2 (3) 5
Quick Check
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Translations
PRE-ALGEBRA LESSON 9-8
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Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
A line graph of Danas weight in one month
resembles a horizontal line. Describe the
situation the graph reflects.
Danas weight has stayed the same during the
one-month period.
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Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
(For help, go to Lesson 8-3.)
Graph each line. 1. x 0 2. y 0 3. x
3 4. y 2 5. x 1 6. x y
Check Skills Youll Need
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Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
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Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Identify the lines of symmetry. Tell how many
there are.
a. 
b. 
Quick Check
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Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Graph the image of FG after a reflection over
the x-axis.
Reflect the other endpoint.
Quick Check
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Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Graph the image of FG after a reflection over y
1.
Graph y 1 (in red).
Reflect the other endpoint.
Quick Check
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Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Use a graph of WXY with vertices W(3, 3),
X(2, 0), and Y(0, 2). 1. Graph WXY. How
many lines of symmetry does WXY
have? 2. Give the vertices of W X Y ,
the image of WXY after a reflection over the
x-axis. 3. Give the vertices of W X Y
, the image of WXY after a reflection over
the y-axis.
1
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Rotations
PRE-ALGEBRA LESSON 9-10
A rectangular field is 120 yd long and 53 yd 1 ft
wide. How much longer is the field than it is
wide?
66 yd 2 ft
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Rotations
PRE-ALGEBRA LESSON 9-10
(For help, go to Lesson 1-10.)
Graph each triangle. 1. A(1, 3), B(4, 1), C(3,
2) 2. J(2, 1), K(1, 3), L(1, 4) 3. X(4, 0),
Y(0, 2), Z(2, 3)
Check Skills Youll Need
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Rotations
PRE-ALGEBRA LESSON 9-10
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Rotations
PRE-ALGEBRA LESSON 9-10
Find the vertices of the image of RST after
a rotation of 90 about the origin.
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Rotations
PRE-ALGEBRA LESSON 9-10
(continued)
Step 2  Rotate the tracing 90 counterclockwise.
Make sure the axes line up. Label the vertices
R , S , and T . Connect the vertices of the
rotated triangle.
Quick Check
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Rotations
PRE-ALGEBRA LESSON 9-10
Judging from appearance, tell whether the star
has rotational symmetry. If so, what is the angle
of rotation?
The star can match itself in 6 positions.
The pattern repeats in 6 equal intervals. 360
6 60
The figure has rotational symmetry.
The angle of rotation is 60.
Quick Check
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Rotations
PRE-ALGEBRA LESSON 9-10
No you cannot rotate the figure 180 or less so
that its image matches the original figure.
yes 45
9-10