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

9-1

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

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.

9-1

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

9-1

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

9-1

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

9-1

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

9-1

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

9-2

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

9-2

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

9-2

Angle Relationships and Parallel Lines

PRE-ALGEBRA LESSON 9-2

Find the measure of 3 if m 4 110.

Quick Check

9-2

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

9-2

Angle Relationships and Parallel Lines

PRE-ALGEBRA LESSON 9-2

145

9-2

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.

9-3

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

9-3

Classifying Polygons

PRE-ALGEBRA LESSON 9-3

Solutions 1. acute 2. obtuse 3. obtuse 4. ri

ght 5. acute 6. obtuse

9-3

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

9-3

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

9-3

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.

9-3

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.

9-3

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.

9-4

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

9-4

Problem Solving Strategy Draw a Diagram

PRE-ALGEBRA LESSON 9-4

9-4

Problem Solving Strategy Draw a Diagram

PRE-ALGEBRA LESSON 9-4

How many diagonals does a nonagon have?

9-4

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

9-4

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

9-4

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

9-5

Congruence

PRE-ALGEBRA LESSON 9-5

(For help, go to Lesson 6-3.)

Check Skills Youll Need

9-5

Congruence

PRE-ALGEBRA LESSON 9-5

9-5

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

9-5

Congruence

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

Congruence

PRE-ALGEBRA LESSON 9-5

(continued)

b.

Quick Check

9-5

Congruence

PRE-ALGEBRA LESSON 9-5

SAS

9-5

Circles

PRE-ALGEBRA LESSON 9-6

Solve the proportion

9-6

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

9-6

Circles

PRE-ALGEBRA LESSON 9-6

Solutions 1. 36 2. 270 3. 54

4. 109

9-6

Circles

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

Circles

PRE-ALGEBRA LESSON 9-6

Make a circle graph for Jackies weekly budget.

9-6

Circles

PRE-ALGEBRA LESSON 9-6

(continued)

Quick Check

9-6

Circles

PRE-ALGEBRA LESSON 9-6

Draw a circle graph of the data.

9-6

Circles

PRE-ALGEBRA LESSON 9-6

(continued)

Use a compass to draw a circle.

Quick Check

9-6

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

9-6

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

9-7

Constructions

PRE-ALGEBRA LESSON 9-7

(For help, go to Lesson 9-1.)

Check Skills Youll Need

9-7

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

9-7

Constructions

PRE-ALGEBRA LESSON 9-7

Construct a segment congruent to WX.

9-7

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

9-7

Constructions

PRE-ALGEBRA LESSON 9-7

Construct an angle congruent to W.

9-7

Constructions

PRE-ALGEBRA LESSON 9-7

(continued)

Quick Check

9-7

Constructions

PRE-ALGEBRA LESSON 9-7

Construct the perpendicular bisector of WY.

9-7

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

9-7

Constructions

PRE-ALGEBRA LESSON 9-7

Construct the bisector of W.

9-7

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

9-7

Constructions

PRE-ALGEBRA LESSON 9-7

13. Check students work.

9-7

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

9-8

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

9-8

Translations

PRE-ALGEBRA LESSON 9-8

9-8

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

9-8

Translations

PRE-ALGEBRA LESSON 9-8

Use arrow notation to describe the

translation of X to X .

Quick Check

9-8

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

9-8

Translations

PRE-ALGEBRA LESSON 9-8

9-8

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.

9-9

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

9-9

Symmetry and Reflections

PRE-ALGEBRA LESSON 9-9

9-9

Symmetry and Reflections

PRE-ALGEBRA LESSON 9-9

Identify the lines of symmetry. Tell how many

there are.

a.

b.

Quick Check

9-9

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

9-9

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

9-9

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

9-9

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

9-10

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

9-10

Rotations

PRE-ALGEBRA LESSON 9-10

9-10

Rotations

PRE-ALGEBRA LESSON 9-10

Find the vertices of the image of RST after

a rotation of 90 about the origin.

9-10

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

9-10

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

9-10

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