Title: Introduction to Geometry: Points, Lines, and Planes
1Introduction 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
9-1
2Introduction 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
3Introduction 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.
9-1
4Introduction 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
9-1
5Introduction 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
9-1
6Introduction 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
9-1
7Introduction 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
9-1
8Angle 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
9-2
9Angle 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
9-2
10Angle 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
9-2
11Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
Find the measure of 3 if m 4 110.
Quick Check
9-2
12Angle 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
9-2
13Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
145
9-2
14Classifying 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.
9-3
15Classifying 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
9-3
16Classifying Polygons
PRE-ALGEBRA LESSON 9-3
Solutions 1. acute 2. obtuse 3. obtuse 4. ri
ght 5. acute 6. obtuse
9-3
17Classifying 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
9-3
18Classifying 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
9-3
19Classifying 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)Â Â Substitute 3 for x.
36 Simplify.
Quick Check
For a side length of 3 ft, the perimeter is 36 ft.
9-3
20Classifying 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.
9-3
21Problem 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.
9-4
22Problem 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
9-4
23Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
9-4
24Problem Solving Strategy Draw a Diagram
PRE-ALGEBRA LESSON 9-4
How many diagonals does a nonagon have?
9-4
25Problem 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
9-4
26Problem 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
9-4
27Congruence
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
9-5
28Congruence
PRE-ALGEBRA LESSON 9-5
(For help, go to Lesson 6-3.)
Check Skills Youll Need
9-5
29Congruence
PRE-ALGEBRA LESSON 9-5
9-5
30Congruence
PRE-ALGEBRA LESSON 9-5
In the figure, TUV WUX.
a. Name the corresponding congruent angles.
b. Name the corresponding congruent sides.
Quick Check
9-5
31Congruence
PRE-ALGEBRA LESSON 9-5
List the congruent corresponding parts of each
pair of triangles. Write a congruence statement
for the triangles.
a.
9-5
32Congruence
PRE-ALGEBRA LESSON 9-5
(continued)
b.
Quick Check
9-5
33Congruence
PRE-ALGEBRA LESSON 9-5
SAS
9-5
34Circles
PRE-ALGEBRA LESSON 9-6
Solve the proportion
9-6
35Circles
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
9-6
36Circles
PRE-ALGEBRA LESSON 9-6
Solutions 1. 36 2. 270 3. 54
4. 109
9-6
37Circles
PRE-ALGEBRA LESSON 9-6
Find the circumference of the circle.
The circumference of the circle is about 37.68 in.
Quick Check
9-6
38Circles
PRE-ALGEBRA LESSON 9-6
Make a circle graph for Jackies weekly budget.
9-6
39Circles
PRE-ALGEBRA LESSON 9-6
(continued)
Quick Check
9-6
40Circles
PRE-ALGEBRA LESSON 9-6
Draw a circle graph of the data.
9-6
41Circles
PRE-ALGEBRA LESSON 9-6
(continued)
Use a compass to draw a circle.
Quick Check
9-6
42Circles
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
9-6
43Constructions
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
9-7
44Constructions
PRE-ALGEBRA LESSON 9-7
(For help, go to Lesson 9-1.)
Check Skills Youll Need
9-7
45Constructions
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
9-7
46Constructions
PRE-ALGEBRA LESSON 9-7
Construct a segment congruent to WX.
9-7
47Constructions
PRE-ALGEBRA LESSON 9-7
(continued)
Step 3Â 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
9-7
48Constructions
PRE-ALGEBRA LESSON 9-7
Construct an angle congruent to W.
9-7
49Constructions
PRE-ALGEBRA LESSON 9-7
(continued)
Quick Check
9-7
50Constructions
PRE-ALGEBRA LESSON 9-7
Construct the perpendicular bisector of WY.
9-7
51Constructions
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
9-7
52Constructions
PRE-ALGEBRA LESSON 9-7
Construct the bisector of W.
9-7
53Constructions
PRE-ALGEBRA LESSON 9-7
(continued)
Step 2Â 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
9-7
54Constructions
PRE-ALGEBRA LESSON 9-7
13. Check students work.
9-7
55Translations
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
9-8
56Translations
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
9-8
57Translations
PRE-ALGEBRA LESSON 9-8
9-8
58Translations
PRE-ALGEBRA LESSON 9-8
Graph the image of BCD after a translation 3
units to the left and 4 units down.
Quick Check
9-8
59Translations
PRE-ALGEBRA LESSON 9-8
Use arrow notation to describe the
translation of X to X .
Quick Check
9-8
60Translations
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
9-8
61Translations
PRE-ALGEBRA LESSON 9-8
9-8
62Symmetry 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.
9-9
63Symmetry 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
9-9
64Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
9-9
65Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Identify the lines of symmetry. Tell how many
there are.
a.Â
b.Â
Quick Check
9-9
66Symmetry 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
9-9
67Symmetry 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
9-9
68Symmetry 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
9-9
69Rotations
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
9-10
70Rotations
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
9-10
71Rotations
PRE-ALGEBRA LESSON 9-10
9-10
72Rotations
PRE-ALGEBRA LESSON 9-10
Find the vertices of the image of RST after
a rotation of 90 about the origin.
9-10
73Rotations
PRE-ALGEBRA LESSON 9-10
(continued)
Step 2Â 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
9-10
74Rotations
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
9-10
75Rotations
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