Transcript No Slide Title

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 You’ll 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
Use the figure to name each of the following.
a. four points
H, I, J, and K
Name a point with a capital letter.
b. four different segments
HO, HJ , KI, and OI
Name a segment by its endpoints.
c. five other names for KI
IK , KO, OK , IO , and OI
Horizontal line KI has several names.
d. five different rays
HO, OJ, KI , OK , and JH
The first letter names the endpoint.
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
MP, OP, QT, ST
b. three segments parallel to PT
MQ, NR, OS
c. four segments skew to PT
MN, NO, QR, RS
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.
First draw two lines that intersect.
Then draw a segment that is parallel to one of
the lines.
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 A, B, C, D
2. another name for EA
3. three different rays
EQ
AQ, AE, GR
4. three segments that are parallel to HG
5. four segments that are skew to CG
6. four segments that intersect AE
9-1
EF, DC, AB
AB, AD, EF, EH
AB, AD, EF, EH
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 You’ll Need
9-2
Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
Solutions
1.
n + 45 = 180
n + 45 – 45 = 180 – 45
n = 135
2.
75 + x = 90
75 – 75 + x = 90 – 75
x = 15
3.
3y = 2y + 90
3y – 2y = 2y – 2y + 90
y = 90
4.
2a + 15 = a + 45
2a – a + 15 = a – a + 45
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
m
m
m
3+m
3 if m
4 = 180°
4 = 110°.
3 and
4 are supplementary.
3 + 110° = 180°
Replace m
4 with 110°.
3 + 110° – 110° = 180° – 110°
Solve for m
3.
m
3 = 70°
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
1
3,
2
4,
5
7,
6
8
b. congruent alternate interior angles
2
7,
6
3
Quick Check
9-2
Angle Relationships and Parallel Lines
PRE-ALGEBRA LESSON 9-2
In the diagram, d e.
1. Find the m 5 if m 8 is 35°.
145°
2. Name the congruent corresponding angles.
1
5,
2
6,
4
8,
3
7
3. Name the congruent alternate interior angles.
4
7,
2
5
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. 160°
4. 90°
5. 36°
6. 127°
Check Skills You’ll Need
9-3
Classifying Polygons
PRE-ALGEBRA LESSON 9-3
Solutions
1. acute
2. obtuse
3. obtuse
4. right
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.
For a side length of 3 ft, the perimeter is 36 ft.
9-3
Quick Check
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
isosceles right triangle
2. a type of quadrilateral that can have opposite sides parallel
and no right angles
parallelogram, rhombus
3. Write a formula for the perimeter of a regular heptagon (7
sides). Evaluate for a side of 12 in.
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.
Sample answer
They are all parallelograms.
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
4. hexagon
5. octagon
3. pentagon
Check Skills You’ll Need
9-4
Problem Solving Strategy: Draw a Diagram
PRE-ALGEBRA LESSON 9-4
Solutions
1.
2.
4.
5.
3.
9-4
Problem Solving Strategy: Draw a Diagram
PRE-ALGEBRA LESSON 9-4
How many diagonals does a nonagon have?
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.
AH, AG, AF, AE, AD, and AC are
some of the diagonals.
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.
A nonagon has 27 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
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 diagonals
2. How many triangles can you form if you draw all the
diagonals from one vertex of a pentagon?
3 triangles
3. How many triangles can you form if you draw all the
diagonals of a rectangle?
8 triangles
9-4
Congruence
PRE-ALGEBRA LESSON 9-5
Replace the question marks with the correct digits.
a. 8 ? 9 + 6. ? ? = 15.96
9, 9, 7
b. 13. ? 0 ? – ? . 4 ? 2 = 4.122
6, 4, 9, 8
9-5
Congruence
PRE-ALGEBRA LESSON 9-5
(For help, go to Lesson 6-3.)
ABC ~ XYZ. For the given part of
part of XYZ.
1.
A
3. AB
2.
ABC, find the corresponding
C
4. CA
Check Skills You’ll Need
9-5
Congruence
PRE-ALGEBRA LESSON 9-5
Solutions
1.
X
3. XY
2.
Z
4. ZX
9-5
Congruence
PRE-ALGEBRA LESSON 9-5
In the figure,
TUV
WUX.
a. Name the corresponding congruent angles.
X, T
W , TUV
WUX
V
b. Name the corresponding congruent sides.
TV
WX , TU
WU , VU
XU
c. Find the length of WX.
Since WX,
TV, and TV = 300 m, WX = 300 m.
9-5
Quick Check
Congruence
PRE-ALGEBRA LESSON 9-5
List the congruent corresponding parts of each pair of
triangles. Write a congruence statement for the triangles.
a.
ACB
AC
ECD
EC
Angle
Side
CAB
CED
ACB
ECD by ASA.
9-5
Angle
Congruence
PRE-ALGEBRA LESSON 9-5
(continued)
b.
MK
MKJ
JK
MKJ
LJ
LJK
JK
Side
Angle
Side
LJK by SAS.
Quick Check
9-5
Congruence
PRE-ALGEBRA LESSON 9-5
Given that
1.
L
O
JKL
MNO, complete the following.
2. JK
MN
3. JL
MO
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?
SAS
9-5
Circles
PRE-ALGEBRA LESSON 9-6
Solve the proportion:
4
n
=
5
27
n = 21 3
5
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.
10
x
=
100
360
2.
75
x
=
100
360
3.
0.8
x
=
5.3
360
4.
1.6
x
=
5.3
360
Check Skills You’ll 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.
C=
C
d
Write the formula.
(3.14)(2)6
Replace
= 37.68
with 3.14 and d with (2)6.
Simplify.
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 Jackie’s weekly budget.
Jackie’s Weekly Budget
Entertainment (e)
20%
Food (f)
20%
Transportation (t)
10%
Savings (s)
50%
Use proportions to find the measures
of the central angles.
20
e
=
100
360
20
f
=
100
360
e = 72°
f = 72°
10
t
=
100
360
50
s
=
100
360
t = 36°
9-6
s = 180°
Circles
PRE-ALGEBRA LESSON 9-6
(continued)
Use a compass to draw a circle.
Draw the central angles with a protractor.
Label each section.
Add a title.
Jackie’s Weekly Budget
Entertainment
Savings
Food
Transportation
9-6
Quick Check
Circles
PRE-ALGEBRA LESSON 9-6
Draw a circle graph of the data.
First add to find the total number of students.
Spring Dance Attendance
Freshmen (f)
Sophomores (p)
120
82
Juniors (j)
137
Seniors (s)
101
120 + 82 + 137 + 101 = 440
Use proportions to find the measures
of the central angles.
120
f
=
440
360
f
98°
137
j
=
440
360
j
9-6
112°
82
p
=
440
360
p
67°
101
s
=
440
360
s
83°
Circles
PRE-ALGEBRA LESSON 9-6
(continued)
Use a compass to draw a circle.
Draw the central angles with a protractor.
Label each section.
Add a title.
Spring Dance Attendance
Seniors
Juniors
Freshmen
Sophomores
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.
about 7.85 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.
about 164°
3. Draw a circle graph of the data.
After-School Number of
Activities
(for one class)
Band
Basketball
Baby-sitting
Library
9-6
Students
5
8
10
7
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.)
State the meaning of each symbol.
1. B
2. AB
3. AB
4. AB
Check Skills You’ll 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.
Step 1 Draw a ray with endpoint G.
Step 2 Open the compass to the
length of 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.
Step 1 Draw a ray with
endpoint A.
Step 2 With the compass
point at W, draw an
arc that intersects
the sides of W.
Label the intersection
points M and N.
9-7
Constructions
PRE-ALGEBRA LESSON 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.
CAB
Step 4 Open the compass to the
length of MN. Using this
setting, put the compass
tip at B. Draw an arc to
determine the point C.
Draw AC.
Quick Check
NWM
9-7
Constructions
PRE-ALGEBRA LESSON 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
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.
ST is perpendicular to WY and ST bisects WY.
Quick Check
9-7
Constructions
PRE-ALGEBRA LESSON 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
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.
WZ bisects
W.
Quick Check
9-7
Constructions
PRE-ALGEBRA LESSON 9-7
Draw and construct the figures.
1. Draw QR; then construct ST
QR.
2. Draw
UVW
LMN; then construct
3. Construct the bisectors of QR and
Questions 1 and 2.
1–3. Check students’ work.
9-7
LMN.
LMN that you drew for
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)
4. D(4, –2)
5. E(–2, –3)
3. C(1, 4)
Check Skills You’ll Need
9-8
Translations
PRE-ALGEBRA LESSON 9-8
Solutions
1–5.
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 .
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
Translations
PRE-ALGEBRA LESSON 9-8
to
Write a rule to describe the translation of
RST.
RST
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
Translations
PRE-ALGEBRA LESSON 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 .
E(1, 1)
E (–1, –3)
3. Write a rule to describe the translation of
(x, y)
(x – 2, y – 4)
9-8
EFG to
EFG.
Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
A line graph of Dana’s weight in one month resembles a horizontal
line. Describe the situation the graph reflects.
Dana’s 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 You’ll Need
9-9
Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Solutions
1.
2.
3.
4.
5.
6.
9-9
Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Identify the lines of symmetry. Tell how many there are.
a.
8 lines of symmetry
b.
2 lines of symmetry
Quick Check
9-9
Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Graph the image of FG after a reflection over the x-axis.
Since F is 2 units below the x-axis,
F is 2 units above the x-axis.
Reflect the other endpoint.
Draw F G .
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).
Since F is 1 unit below the red line,
F is 1 unit above the red line.
Reflect the other endpoint.
Draw F G .
Quick Check
9-9
Symmetry and Reflections
PRE-ALGEBRA LESSON 9-9
Use a graph of
1. Graph
have?
WXY with vertices W(–3, 3), X(–2, 0), and Y(0, 2).
WXY. How many lines of symmetry does
WXY
1
2. Give the vertices of W X Y , the image of
reflection over the x-axis.
WXY after a
W (–3, –3), X (–2, 0), Y (0, –2)
3. Give the vertices of W X Y , the image of
reflection over the y-axis.
W (3, 3), X (2, 0), Y (0, 2)
9-9
WXY after a
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 You’ll Need
9-10
Rotations
PRE-ALGEBRA LESSON 9-10
Solutions
1.
2.
3.
9-10
Rotations
PRE-ALGEBRA LESSON 9-10
Find the vertices of the image of
90° about the origin.
Step 1
RST after a rotation of
Use a blank transparency sheet.
Trace RST, the x-axis, and the y-axis.
Then fix the tracing in place at 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.
The vertices of the image are
R (1, 1), S (4, 1), and T (4, 5).
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
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.
D (2, –4), F (3, –1), G (1, –2)
Judging from appearance, tell whether each figure has rotational
symmetry. If so, what is the angle of rotation? If not, explain.
2.
3.
yes; 45°
No; you cannot rotate the figure 180°
or less so that its image matches the
original figure.
9-10