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Transcript
```Pairs of Angles
LESSON 7-1
Course 3
Problem of the Day
Students at West Ridge Middle School are going on a trip to the museum.
Nine have never gone before. Twelve have gone once, half as many as that
have gone twice, and none have gone more than that. How many students
are going on the trip?
27
Lesson
Main
Lesson
7-1
Feature
Pairs of Angles
LESSON 7-1
Course 3
Check Skills You’ll Need
(For help, go to Lesson 1-6.)
1. Vocabulary Review What is the inverse operation of addition?
Solve each equation.
2. a + 14 = 32
3. b – 5 = 26
4. 10 + c = –31
5. –48 = d – 19
Check Skills You’ll Need
Lesson
Main
Lesson
7-1
Feature
Pairs of Angles
LESSON 7-1
Course 3
Check Skills You’ll Need
Solutions
1. subtraction
2. 18
3. 31
4. –41
5. –29
Lesson
Main
Lesson
7-1
Feature
Pairs of Angles
LESSON 7-1
Course 3
Find the measure of the supplement of
x° + m
IGJ = 180°
IGJ.
The sum of the measures of
supplementary angles is 180º.
x° + 145° = 180°
Substitute 145º for m DEF.
x° + 145° – 145° = 180° – 145°
Subtract 145º from each side.
x° = 35°
Simplify.
The measure of the supplement of m IGJ is 35º.
Lesson
Main
Lesson
7-1
Quick Check
Feature
Pairs of Angles
LESSON 7-1
Course 3
Name a pair of adjacent angles and a pair of
vertical angles in the figure. Find m HGK.
JGI and
IGH;
HGK and
IGH and
The vertical angles are
KGJ;
KGJ and
JGI;
HGK.
JGI and
HGK;
Since vertical angles are congruent, m
HGI and
HGK = m
KGJ.
JGI = 145°.
Quick Check
Lesson
Main
Lesson
7-1
Feature
Pairs of Angles
LESSON 7-1
Course 3
In this figure, if m
GKJ and JKF.
of
m
DKE + 90° = 180°
m
DKE = 90°
Lesson
Main
DKH = 73°, find the measures
DKE and
FKE are supplementary.
Subtract 90º from each side.
Lesson
7-1
Feature
Pairs of Angles
LESSON 7-1
Course 3
(continued)
m
KHE and
KHE + 73° = 90°
DKH are complementary.
m
KHE = 17°
m
GKJ = m
KHE = 17°
GKJ and
KHE are vertical angles.
m
JKF = m
DKH = 73°
JKF and
DKH are vertical angles.
So, the measure of
Subtract 73º from each side.
GKJ is 17° is and the measure of
JKF is 73°.
Quick Check
Lesson
Main
Lesson
7-1
Feature
Pairs of Angles
LESSON 7-1
Course 3
Lesson Quiz
Use the diagram to answer Questions 1 – 3.
1. List all pairs of vertical angles.
AXD and BXC; AXB and
2. List any angles adjacent to
AXD and BXC
DXC
CXD.
3. If m AXB = 110°, find m DXC.
110°
4. An angle measures 57°. What is the measure of its supplement?
123°
Lesson
Main
Lesson
7-1
Feature
Angles and Parallel Lines
LESSON 7-2
Course 3
Problem of the Day
Write the following sentence in mathematical symbols. Then indicate whether
the sentence is true or false: Four and one sixth minus five eighths equals
three and thirteen twenty fourths.
1
5
13
4 6 – 8 = 3 24 ; true
Lesson
Main
Lesson
7-2
Feature
Angles and Parallel Lines
LESSON 7-2
Course 3
Check Skills You’ll Need
(For help, go to Lesson 7-1.)
1. Vocabulary Review Which of the following pairs of
angles are supplementary?
50° and 40°, 100° and 90°, 120° and 60°, 75° and 125°
Find the measure of the supplement of each angle.
2. 48°
3. 119°
4. 67°
5. 131°
Check Skills You’ll Need
Lesson
Main
Lesson
7-2
Feature
Angles and Parallel Lines
LESSON 7-2
Course 3
Check Skills You’ll Need
Solutions
1. 120° and 60°
2. 132°
3. 61°
4. 113°
5. 49°
Lesson
Main
Lesson
7-2
Feature
Angles and Parallel Lines
LESSON 7-2
Course 3
Identify each pair of corresponding angles and
each pair of alternate interior angles.
1 and
angles.
3,
2 and
4,
5 and
7,
6 and
8 are pairs of corresponding
2 and
7,
3 and
6, are pairs of alternate interior angles.
Quick Check
Lesson
Main
Lesson
7-2
Feature
Angles and Parallel Lines
LESSON 7-2
Course 3
If p is parallel to q, and m 3 = 56º, find m 6.
m
6=m
m
6 = 56°
3 = 56°
Alternate interior angles are congruent.
Quick Check
Lesson
Main
Lesson
7-2
Feature
Angles and Parallel Lines
LESSON 7-2
Course 3
In the diagram below, m 5 = m 6 = and m 7 = 80º. Explain
why p and q are parallel and why s and t are parallel.
p || q because
5 and
7 are congruent alternate interior angles.
s || t because
6 and
7 are congruent corresponding angles.
Quick Check
Lesson
Main
Lesson
7-2
Feature
Angles and Parallel Lines
LESSON 7-2
Course 3
Lesson Quiz
Use the diagram to answer the questions.
1. Classify 4 and 7 as
alternate interior angles,
corresponding angles,
or neither.
neither
2. Classify 2 and 8 as
alternate interior angles,
corresponding angles,
or neither.
alternate interior angles
3. If a || b and m 8 = 80°,
find m 4.
4. Suppose m 5 = 100° and
m 3 = 100°. What can you
line b?
a || b
80°
Lesson
Main
Lesson
7-2
Feature
Congruent Polygons
LESSON 7-3
Course 3
Problem of the Day
Solve the proportion:
4
n
= .
5 27
3
n = 21 5
Lesson
Main
Lesson
7-3
Feature
Congruent Polygons
LESSON 7-3
Course 3
Check Skills You’ll Need
(For help, go to Lesson 4-4.)
1. Vocabulary Review Congruent angles have ? measures.
Are the polygons similar? Explain.
2.
Check Skills You’ll Need
Lesson
Main
Lesson
7-3
Feature
Congruent Polygons
LESSON 7-3
Course 3
Check Skills You’ll Need
Solutions
1. equal
2. LN
ZY
LM ; 5 =/ 4 ; not similar; corresponding sides are not in
XZ 15 10
proportion.
Lesson
Main
Lesson
7-3
Feature
Congruent Polygons
LESSON 7-3
Course 3
In the diagram below, list the congruent parts of
the two figures. Then write a congruence statement.
Congruent Angles
A
M
B
N
C
O
D
P
Congruent Sides
AB MN
BC NO
CD OP
DA PM
Since A corresponds to M, B corresponds to N, C corresponds to
O, and D corresponds to P, a congruence statement is ABCD MNOP.
Quick Check
Lesson
Main
Lesson
7-3
Feature
Congruent Polygons
LESSON 7-3
Course 3
Show that each pair of triangles is congruent.
a.
b.
Q
QP
Q
QPR
E
EY
Y
Angle
Side
Angle
EYT by ASA.
SQ
VT
Q
T
QR
TU
SQR
Side
Angle
Side
VTU by SAS.
Quick Check
Lesson
Main
Lesson
7-3
Feature
Congruent Polygons
LESSON 7-3
Course 3
A surveyor drew the diagram below to find the distance from
J to I across the canyon. Show that GHI
KJI. Then find JK.
J
JI
KIJ
H
Both are right angles.
HI
Both measure 48 ft.
GIH
They are vertical angles.
So ∆GHI
∆ JI by ASA
Corresponding parts of congruent triangles are congruent.
JK corresponds to HG, so JK is 36 ft.
Lesson
Main
Lesson
7-3
Quick Check
Feature
Congruent Polygons
LESSON 7-3
Course 3
Lesson Quiz
Use
ABC and
1.Suppose AC= XZ, AB = XY, and BC = YZ.
Write a congruence statement for the figures.
∆ABC  ∆XYZ
2. Suppose ABC and XYZ are congruent. If AB = 5 cm,
BC = 8 cm, and AC = 10 cm, find XZ.
10 cm
3. Suppose
ABC
B
Y,
XYZ?
A
X, and AB
XY. Why is
ASA
Lesson
Main
Lesson
7-3
Feature
Congruent Polygons
LESSON 7-3
Course 3
Lesson Quiz
4. Let AB = XY = 9 inches; BC = YZ = 24 inches; and
m B = 85°, m Z = 35°, and m Y = 85°. Prove that
the triangles are congruent and find m C.
∆ABC  ∆XYZ by SAS; m C = 35°
Lesson
Main
Lesson
7-3
Feature
LESSON 7-4
Course 3
Problem of the Day
Ruth was born on February 29, 1984. If she insists on celebrating her
birthday only on February 29, when will she celebrate her 12th birthday?
2032
Lesson
Main
Lesson
7-4
Feature
LESSON 7-4
Course 3
Check Skills You’ll Need
(For help, go to the Skills Handbook page 640.)
1. Vocabulary Review How many degrees does a right
angle have?
Classify each angle as acute, right, obtuse, or straight.
2.
3.
Check Skills You’ll Need
Lesson
Main
Lesson
7-4
Feature
LESSON 7-4
Course 3
Check Skills You’ll Need
Solutions
1. 90°
2. acute
3. obtuse
Lesson
Main
Lesson
7-4
Feature
LESSON 7-4
Course 3
Classify
LMN by its sides and angles.
The triangle has two sides that are congruent and three acute angles.
It is an isosceles acute triangle.
Quick Check
Lesson
Main
Lesson
7-4
Feature
LESSON 7-4
Course 3
What is the best name for figure WXYZ? Explain
WXYZ has both pairs of opposite sides parallel, but adjacent sides
are not equal, so it is a parallelogram.
Quick Check
Lesson
Main
Lesson
7-4
Feature
LESSON 7-4
Course 3
Lesson Quiz
1. Classify the triangle according to its angles and sides.
obtuse isosceles triangles
2. A triangle’s sides are all congruent and its angles all
measure 60°. Classify the triangle.
equilateral, acute
3. Determine the best name for the quadrilateral.
rhombus
Lesson
Main
Lesson
7-4
Feature
LESSON 7-4
Course 3
Lesson Quiz
4. What is the best name for a figure that has four sides
congruent, corresponding sides parallel, and all four
angles congruent?
square
Lesson
Main
Lesson
7-4
Feature
Angles and Polygons
LESSON 7-5
Course 3
Problem of the Day
A pound of turkey has 144 g of protein and will serve 4 people. If 4 people
are served equal amounts, how many grams of protein will each receive?
36 g
Lesson
Main
Lesson
7-5
Feature
Angles and Polygons
LESSON 7-5
Course 3
Check Skills You’ll Need
(For help, go to the Lesson 1-1.)
1. Vocabulary Review How do you evaluate an algebraic
expression?
Evaluate each expression for a = 8.
2. 3(a + 1)
3. 5a + 8
a
4. (a – 2)6
Check Skills You’ll Need
Lesson
Main
Lesson
7-5
Feature
Angles and Polygons
LESSON 7-5
Course 3
Check Skills You’ll Need
Solutions
1. You replace each variable in the expression with a number and then
simplify.
2. 27
3. 6
4. 36
Lesson
Main
Lesson
7-5
Feature
Angles and Polygons
LESSON 7-5
Course 3
Find the sum of the measures of the interior angles
of an octagon.
An octagon has 8 sides.
(n – 2) 180º = (8 – 2) 180º
Substitute 8 for n.
= (6) 180º
Subtract.
= 1,080º
Simplify.
The sum of the interior angles of an octagon is 1,080.
Quick Check
Lesson
Main
Lesson
7-5
Feature
Angles and Polygons
LESSON 7-5
Course 3
Find the missing angle measure in the hexagon.
Step 1 Find the sum of the measures of the interior angles of a hexagon.
(n – 2) 180° = (6 – 2) 180°
Substitute 6 for n.
= 720°
Lesson
Main
Simplify.
Lesson
7-5
Feature
Angles and Polygons
LESSON 7-5
Course 3
(continued)
Step 2 Write an equation.
Let x = the missing angle measure.
720° = 120° + 115° + 136° + 80° + 147° + x°
Write an equation.
720° = 598° + x°
122° = x°
Subtract 598º from
each side.
The missing angle measure is 122º.
Quick Check
Lesson
Main
Lesson
7-5
Feature
Angles and Polygons
LESSON 7-5
Course 3
A design on a tile is in the shape of a regular nonagon. Find
the measure of each angle.
(n – 2) 180° = (9 – 2) 180°
Substitute 9 for n since a
nonagon has 9 sides.
Simplify.
= 1,260°
Divide the sum by the number
of angles in a nonagon.
1,260° ÷ 9 = 140°
Each angle of a regular nonagon has a measure of 140°.
Quick Check
Lesson
Main
Lesson
7-5
Feature
Angles and Polygons
LESSON 7-5
Course 3
Lesson Quiz
1. Find the sum of the measures of the interior angles of a polygon having
17 sides.
2,700°
2. Five angles of a hexagon measure 128°, 190°, 112°, 154°, and 90°.
Find the measure of the missing angle.
46°
3. Find the measure of each angle of a regular polygon having 24 sides.
165°
4. A regular figure has an interior angle measure of 135°.
How many sides does it have?
8
Lesson
Main
Lesson
7-5
Feature
Areas of Polygons
LESSON 7-6
Course 3
Problem of the Day
Lucy predicted that her final average for math class would be at least a 93.
Her test grades were 88, 90, 92, 97. What is the lowest test score she can
make on the last test to make this true?
98
Lesson
Main
Lesson
7-6
Feature
Areas of Polygons
LESSON 7-6
Course 3
Check Skills You’ll Need
(For help, go to Lesson 2-6.)
1. Vocabulary Review What is a formula?
Find the area of each figure.
2.
3.
Check Skills You’ll Need
Lesson
Main
Lesson
7-6
Feature
Areas of Polygons
LESSON 7-6
Course 3
Check Skills You’ll Need
Solutions
1. A formula is a rule that shows the relationship between two or more
quantities.
2. A = • w
= 10 • 8
A = 80 cm2
Lesson
Main
3. A = s2
= 72
A = 49 ft2
Lesson
7-6
Feature
Areas of Polygons
LESSON 7-6
Course 3
Find the area of the triangular part of the doghouse.
1
A = 2 bh
Use the area of a triangle formula.
1
= 2 • 36 • 21
Substitute 36 for b and 21 for h.
= 378
Multiply.
The area of the triangular part of the doghouse is 378 in.2.
Quick Check
Lesson
Main
Lesson
7-6
Feature
Areas of Polygons
LESSON 7-6
Course 3
Find the area of the trapezoid.
A = 1 h (b1 + b2)
2
Use the formula.
= 1 (4.4) (6.7 + 9.3)
Substitute 4.4 for h, 6.7 for b1, and 9.3 for b2.
= 35.2
Simplify.
2
The area of the trapezoid is 35.2 in.2.
Lesson
Main
Lesson
7-6
Quick Check
Feature
Areas of Polygons
LESSON 7-6
Course 3
Lesson Quiz
1. Find the area.
14 cm2
2. An architect is designing a restaurant with a triangular
entrance. The base of the triangle is 10 ft wide. The
entrance is 14 ft tall. Find the area of the triangle.
70 ft2
Lesson
Main
Lesson
7-6
Feature
Areas of Polygons
LESSON 7-6
Course 3
Lesson Quiz
3. Find the area.
56 ft2
4. If both bases of the figure in Exercise 3 are doubled, what is the
new area of the trapezoid?
112 ft2
Lesson
Main
Lesson
7-6
Feature
Circumference and Area of a Circle
LESSON 7-7
Course 3
Problem of the Day
Opal, Charles, Jean, and Scott had an earthworm-catching contest. Jean
caught one fourth as many worms as Opal and twice as many as Charles.
Opal caught 3 times as many worms as Scott. How many worms did each
of the other three contestants catch if Scott caught 8 worms?
Opal, 24; Jean, 6; Charles, 3
Lesson
Main
Lesson
7-7
Feature
Circumference and Area of a Circle
LESSON 7-7
Course 3
Check Skills You’ll Need
(For help, go to Lesson 7-6.)
1. Vocabulary Review Explain the difference between
perimeter and area.
Find the area of the figure below.
2.
Check Skills You’ll Need
Lesson
Main
Lesson
7-7
Feature
Circumference and Area of a Circle
LESSON 7-7
Course 3
Check Skills You’ll Need
Solutions
1. Perimeter is the distance around a figure. Area is the number of square
units a figure encloses.
2. A
=
1
bh
2
= 1 • 13 • 7
2
A = 45.5 ft²
Lesson
Main
Lesson
7-7
Feature
Circumferences and Areas of Circles
LESSON 7-7
Course 3
The diameter of a tractor tire is 125 cm. Find the
circumference and area. Round to the nearest tenth.
C=
=
d
Use the formula for circumference.
(125)
Substitute 125 for d.
125
Use a calculator.
392.6990817
The circumference is about 392.7 cm.
Lesson
Main
Lesson
7-7
Feature
Circumferences and Areas of Circles
LESSON 7-7
Course 3
(continued)
A=
=
r2
Use the formula for the area of a
circle.
(62.5) 2
The radius is 125 ÷ 2, or
62.5. Substitute 62.5 for r.
62.5
Use a calculator.
12271.8463
The area is about 12,271.8 cm2.
Quick Check
Lesson
Main
Lesson
7-7
Feature
Circumferences and Areas of Circles
LESSON 7-7
Course 3
Find the area of the unshaded region of the square tile with a
circle inside of it, as shown below. Round to the nearest tenth.
You can separate the figure into a circle and a square.
Lesson
Main
Lesson
7-7
Feature
Circumferences and Areas of Circles
LESSON 7-7
Course 3
(continued)
Step 1 Find the area of the square.
Area of square = s2
A = (12)2
Substitute 12 for s.
= 144
Simplify.
Step 2 Find the area of the circle.
Area of circle =
A=
r2
(6)2
113.1
Lesson
Main
Substitute 6 for r.
Multiply. Round to the nearest tenth.
Lesson
7-7
Feature
Circumferences and Areas of Circles
LESSON 7-7
Course 3
(continued)
Step 3 Subtract the area of the circle from the area of the square.
144 cm2 – 113.1 cm2 = 30.9 cm2.
Quick Check
Lesson
Main
Lesson
7-7
Feature
Circumferences and Areas of Circles
LESSON 7-7
Course 3
Lesson Quiz
1. Find the circumference and area of the circle. Round to the nearest
tenth.
18.8 km; 28.3 km2
2. The diameter of the lid of a can of soup is 5 cm. Find its
circumference and area. Round to the nearest tenth.
15.7 cm; 19.6 cm2
Lesson
Main
Lesson
7-7
Feature
Circumferences and Areas of Circles
LESSON 7-7
Course 3
Lesson Quiz
3. Find the area. Round to the nearest tenth.
273.0 in.2
Lesson
Main
Lesson
7-7
Feature
Constructions
LESSON 7-8
Course 3
Problem of the Day
Estimate in years the age of someone who is a million minutes old.
Lesson
Main
Lesson
7-8
Feature
Constructions
LESSON 7-8
Course 3
Check Skills You’ll Need
(For help, go to Lesson 7-3.)
1. Vocabulary Review How are congruent polygons
the same?
List the congruent parts of each pair of congruent figures.
2. ∆PQR
∆TUV
3. ABCD
LMNO
Check Skills You’ll Need
Lesson
Main
Lesson
7-8
Feature
Constructions
LESSON 7-8
Course 3
Check Skills You’ll Need
Solutions
1. Congruent polygons have the same size and shape.
2. P
PQ
3. A
AB
T;
Q
TU; QR
L;
B
LM; BC
Lesson
Main
U; R
V;
UV; RP
VT
M; C
MN; CD
N; D
NO; DA
Lesson
7-8
O;
OL
Feature
Constructions
LESSON 7-8
Course 3
Use a protractor to draw a 40º angle and label it
Construct N congruent to B.
B.
Step 1 Draw a ray with endpoint N.
Step 2 Put the compass tip at B and
draw an arc that intersects the
sides of B. Label the points
of intersection A and C.
Step 3 Keep the compass open to the
same width. Put the compass
tip at N. Draw an arc that
intersects the ray at a point O.
Lesson
Main
Lesson
7-8
Feature
Constructions
LESSON 7-8
Course 3
(continued)
Step 4 Adjust the compass width
So that the tip is at C and
the pencil is at A.
Step 5 Keep the compass open to the
same width. Put the compass tip
on O and draw an arc that
intersects the first arc at point M.
Step 6 Draw NM.
N is congruent to
Lesson
Main
Quick Check
B.
Lesson
7-8
Feature
Constructions
LESSON 7-8
Course 3
Draw a line f. Construct a line parallel to line f.
Step 1 Draw line f.
Step 2 Draw line k that intersects line f at D.
Label the angle formed 1. Then label
point E on line k.
Step 3 Construct an angle at E that is congruent to 1.
EC is parallel to line f.
Quick Check
Lesson
Main
Lesson
7-8
Feature
Constructions
LESSON 7-8
Course 3
Lesson Quiz
For Exercises 1–2, use the diagram below.
1. Construct an angle that is congruent to R.
2. Construct a line parallel to RS.
Lesson
Main
Lesson
7-8
Feature
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