Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 9-1 The telephone company...

Introduction to Geometry: Points, Lines, and PlanesIntroduction to Geometry: Points, Lines, and PlanesPRE-ALGEBRA LESSON 9-1PRE-ALGEBRA LESSON 9-1

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


(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 You’ll Need

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

Use the figure to name each of the following.

H, I, J, and K Name a point with a capital letter.


a. four points

b. four different segments

c. five other names for KI

d. five different rays

, and OI, IO, OK, KO

HO Name a segment by its endpoints., HJ, KI, and OI

IK Horizontal line KI has several names.

The first letter names the endpoint.HO, OJ, KI , OK, and JH

Quick Check

9-1

MP, OP, QT, ST

MQ, NR, OS

MN, NO, QR, RS

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

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.

First draw two lines that intersect.

Then draw a segment that is parallel to one of the lines.

Quick Check

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

EQ

AQ, AE, GR

EF, DC, AB

AB, AD, EF, EH

AB, AD, EF, EH

9-1

Angle Relationships and Parallel LinesAngle Relationships and Parallel LinesPRE-ALGEBRA LESSON 9-2PRE-ALGEBRA LESSON 9-2

9-2

The Jackson County Bird Sanctuary has three times as many owls ashawks. It has 40 hawks and owls in all. How many of each are in the sanctuary?

30 owls, 10 hawks



Solve.

1. n + 45 = 180 2. 75 + x = 90

3. 3y = 2y + 90 4. 2a + 15 = a + 45


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

Find the measure of 3 if m 4 = 110°.


Replace m 4 with 110°.m 3 + 110° = 180°

Solve for m 3.m 3 + 110° – 110° = 180° – 110°

m 3 = 70°

3 and 4 are supplementary.m 3 + m 4 = 180°

Quick Check

9-2

1 3, 2 4, 5 7, 6 8

2 7, 6 3

In the diagram, p || q. Identify each of the following.


a. congruent corresponding angles

b. congruent alternate interior angles

Quick Check

9-2


145°

In the diagram, d e.

1. Find the m 5 if m 8 is 35°.

2. Name the congruent corresponding angles.

3. Name the congruent alternate interior angles.

1 5, 2 6, 4 8, 3 7

4 7, 2 5

9-2

Classifying PolygonsClassifying PolygonsPRE-ALGEBRA LESSON 9-3PRE-ALGEBRA LESSON 9-3

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.



For the angle measures given, classify the angle as acute, right, or obtuse.

1. 85° 2. 95° 3. 160°

4. 90° 5. 36° 6. 127°


9-3


Solutions

1. acute 2. obtuse 3. obtuse

4. right 5. acute 6. obtuse

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

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

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.

For a side length of 3 ft, the perimeter is 36 ft.Quick Check

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. P = 7x; 84 in.

isosceles right triangle

parallelogram, rhombus

9-3

Problem Solving Strategy: Draw a DiagramProblem Solving Strategy: Draw a DiagramPRE-ALGEBRA LESSON 9-4PRE-ALGEBRA LESSON 9-4

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.

They are all parallelograms.

Sample answer



Sketch each figure.

1. equilateral triangle 2. rectangle 3. pentagon

4. hexagon 5. octagon


9-4


Solutions

1. 2. 3.

4. 5.

9-4


How many diagonals does a nonagon have?

AH, AG, AF, AE, AD, and AC are some of the diagonals.

One strategy for solving this problem is to draw a diagram and count the diagonals. A nonagon has nine sides. You can draw six diagonals from one vertex of a nonagon.

9-4


(continued)

A nonagon has 27 diagonals.

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 DiagonalsABCDEFGHI

665432100

Total 27

Quick Check

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?

8 triangles

2 diagonals

3 triangles

9-4

CongruenceCongruencePRE-ALGEBRA LESSON 9-5PRE-ALGEBRA LESSON 9-5

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



ABC ~ XYZ. For the given part of ABC, find the corresponding part of XYZ.

1. A 2. C

3. AB 4. CA


9-5


Solutions

1. X 2. Z

3. XY 4. ZX

9-5

V X, T W, TUV WUX

TV WX WU, TU XU, VU


In the figure, TUV WUX.

a. Name the corresponding congruent angles.

b. Name the corresponding congruent sides.

c. Find the length of WX.

WX, TV,Since TV = 300 m, WX = 300 m.andQuick Check

9-5


List the congruent corresponding parts of each pair of triangles. Write a congruence statement for the triangles.

ACB

ECD

Angle

AC

EC

Side

CAB

CED

Angle

ACB

ECD

by ASA.

a.

9-5


(continued)

by SAS.MKJ LJK

MK

LJ

Side

MKJ LJK Angle

JK JK

Side

b.

Quick Check

9-5


Given that JKL MNO, complete the following.

1. L 2. JK 3. JL

4. If two sides and the angle between those sides of one triangle are congruent to two sides and the angle between those sides of another triangle, why can you conclude that the two triangles are congruent?

O MN MO

SAS

9-5

CirclesCirclesPRE-ALGEBRA LESSON 9-6PRE-ALGEBRA LESSON 9-6

9-6

Solve the proportion: =45

n27

35

n = 21



Solve each proportion. Round to the nearest whole number where necessary.

1. = 2. =

3. = 4. =

10100

x360

75100

x360

0.8 5.3

x360

1.6 5.3

x360


9-6


Solutions

1. 36 2. 270

3. 54 4. 109

9-6


Find the circumference of the circle.

C =

d

Write the formula.

= 37.68 Simplify.

C

(3.14)(2)6

Replace with 3.14 and d with (2)6.

The circumference of the circle is about 37.68 in.

Quick Check

9-6


Make a circle graph for Jackie’s weekly budget.

Jackie’s Weekly Budget

Entertainment (e)

Food (f)

Transportation (t)

Savings (s)

20%

20%

10%

50%

Use proportions to find the measures of the central angles.

=

=

e = 72°

t = 36°

t360

10100

e360

20100 =

f = 72°

=

s = 180°

50100

s360

f360

20100

9-6

Use a compass to draw a circle.

Draw the central angles with a protractor.


(continued)

Add a title.

Jackie’s Weekly Budget

Label each section.

Savings

Entertainment

Food

Transportation Quick Check

9-6


Draw a circle graph of the data.

First add to find the total number of students.

120 + 82 + 137 + 101 = 440

p 67°

s 83°

Use proportions to find the measures of the central angles.

f 98°

j 112°

=137440

j360 =

101440

s360

=p

36082

440=120440

f360

Spring Dance Attendance

Freshmen (f)

Sophomores (p)

Juniors (j)

Seniors (s)

120

82

137

101

9-6


(continued)

Use a compass to draw a circle.

Draw the central angles with a protractor.

SeniorsFreshmen

JuniorsSophomores

Label each section.

Spring Dance Attendance

Add a title.

Quick Check

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°

After-School Number of Activities Students(for one class) Band 5 Basketball 8 Baby-sitting 10 Library 7

9-6

ConstructionsConstructionsPRE-ALGEBRA LESSON 9-7PRE-ALGEBRA LESSON 9-7

9-7

A rectangular field is three times as long as it is wide. What are itswidth and length if the perimeter is 600 yd?

width: 75 yd; length: 225 yd



State the meaning of each symbol.

1. B 2. AB 3. AB 4. AB


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


Construct a segment congruent to WX.

Step 1 Draw a ray with endpoint G.

Step 2 Open the compass to the length of WX.

9-7


Step 3 With the same compass setting, put the compass tip on G. Draw an arc that intersects the ray. Label the intersection H.

(continued)

GH WX

Quick Check

9-7


Construct an angle congruent to W.

Step 2 With the compass point at W, draw an arc that intersects the sides of W. Label the intersection points M and N.

Step 1 Draw a ray with endpoint A.

9-7


(continued)

Step 3 With the same compass setting, put the compass tip on A. Draw an arc that intersects the ray at point B.

Step 4 Open the compass to the length of MN. Using thissetting, put the compass tip at B. Draw an arc to determine the point C. Draw AC.

CAB NWM Quick Check

9-7


Construct the perpendicular bisector of WY.

Step 1 Open the compass to more than half the length of WY. Put the compass tip at W. Draw an arc intersecting WY. With the same compass setting, repeat from point Y.

9-7

PRE-ALGEBRA LESSON 9-7PRE-ALGEBRA LESSON 9-7

Step 2 Label the points of intersection S and T. Draw ST. Label the intersection of ST and WY point M.

ST is perpendicular to WY and ST bisects WY.

(continued)

ConstructionsConstructions

Quick Check

9-7


Construct the bisector of W.

Step 1 Put the compass tip at W. Draw an arc that intersects the sides of W. Label the points of intersection S and T.


9-7


(continued)

WZ bisects W.

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


1–3. Check students’ work.

Draw and construct the figures.

1. Draw QR; then construct ST QR.

2. Draw LMN; then construct UVW LMN.

3. Construct the bisectors of QR and LMN that you drew for

Questions 1 and 2.

9-7

TranslationsTranslationsPRE-ALGEBRA LESSON 9-8PRE-ALGEBRA LESSON 9-8

9-8

The short sides of a kite measure 54 cm each, and the longsides each measure 78 cm. What is the perimeter of the kite?

264 cm



Graph each point.

1. A(–4, 3) 2. B(0, 2) 3. C(1, 4)

4. D(4, –2) 5. E(–2, –3)


9-8


Solutions

1–5.

9-8

Graph the image of BCD after a translation3 units to the left and 4 units down.


TranslationsTranslations

Quick Check

9-8

Use arrow notation to describe the translationof X to X .



The point moves from X(–2, 3) to X (3, 1), so the translation is X(–2, 3) X (3, 1).

Quick Check

9-8

Write a rule to describe the translation of RST to R S T .


Use R(–2, –3) and its image R (–3, 2) to find the horizontal and vertical translations.

Horizontal translation: –3 – (–2) = –1

Vertical translation: 2 – (–3) = 5

The rule is (x, y) (x – 1, y + 5).


Quick Check

9-8


For Exercises 1–3, use a translation 2 units to the left and 4 units down.

1. Graph EFG with vertices E(1, 1), F(4, 4), and G(4, 1). Also graph its image, E F

G .

2. Use arrow notation to describe the translation of E to E .

3. Write a rule to describe the translation of EFG to E F G .

E(1, 1) E (–1, –3)

(x, y) (x – 2, y – 4)

9-8

Symmetry and ReflectionsSymmetry and ReflectionsPRE-ALGEBRA LESSON 9-9PRE-ALGEBRA LESSON 9-9

9-9

A line graph of Dana’s weight in one month resembles a horizontalline. Describe the situation the graph reflects.

Dana’s weight has stayed the same during the one-month period.



Graph each line.

1. x = 0 2. y = 0 3. x = 3

4. y = 2 5. x = –1 6. x = y


9-9


Solutions

1. 2. 3.

4. 5. 6.

9-9

Identify the lines of symmetry. Tell how many there are.


a.

b. 2 lines of symmetry

8 lines of symmetry

Symmetry and ReflectionsSymmetry and Reflections

Quick Check

9-9

Graph the image of FG after a reflection over the x-axis.


Reflect the other endpoint.


Since F is 2 units below the x-axis, F is 2 units above the x-axis.

Draw F G .

Quick Check

9-9

Graph the image of FG after a reflection over y = –1.


Graph y = –1 (in red).

Reflect the other endpoint.


Since F is 1 unit below the red line, F is 1 unit above the red line.

Draw F G .

Quick Check

9-9


W (3, 3), X (2, 0), Y (0, 2)

1

W (–3, –3), X (–2, 0), Y (0, –2)

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.

9-9

RotationsRotationsPRE-ALGEBRA LESSON 9-10PRE-ALGEBRA LESSON 9-10

9-10

A rectangular field is 120 yd long and 53 yd 1 ft wide. How muchlonger is the field than it is wide?

66 yd 2 ft



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)


9-10


Solutions

1. 2. 3.

9-10

Find the vertices of the image of RST after a rotation of

90° about the origin.


Step 1 Use a blank transparency sheet.Trace RST, the x-axis, and the y-axis. Then fix the tracing in place at the origin.

RotationsRotations

9-10

(continued)


RotationsRotations

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.

The vertices of the image are R (1, 1), S (4, 1), and T (4, 5).

Quick Check

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°.

RotationsRotations

Quick Check

9-10


No; you cannot rotate the figure 180° or less so that its image matches the original figure.

D (2, –4), F (3, –1), G (1, –2)

yes; 45°

1. DFG has vertices D(–2, 4), F(–3, 1), and G(–1, 2). Find the vertices of the image D F G after a rotation of 180°

about the origin.

Judging from appearance, tell whether each figure has rotational symmetry. If so, what is the angle of rotation? If not, explain.

2. 3.

9-10

Introduction to Geometry: Points, Lines, and Planes PRE-ALGEBRA LESSON 9-1 9-1 The telephone company...

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