Lesson 10-1
Line and Angle Relationships
Lesson 10-2
Congruent Triangles
Lesson 10-3
Transformations on the Coordinate Plane
Lesson 10-4
Quadrilaterals
Lesson 10-5
Area: Parallelograms, Triangles,
and Trapezoids
Lesson 10-6
Polygons
Lesson 10-7
Circumference and Area: Circles
Lesson 10-8
Area: Irregular Figures
Example 1 Find Measures of Angles
Example 2 Find a Missing Angle Measure
Example 3 Find Measures of Angles
Example 4 Apply Angle Relationships
In the figure, m || n and t is a transversal. If
find m2 and m8.
Since
are alternate
exterior angles, they are
congruent. So,
.
Answer:
Since
are
corresponding angles, they are
congruent. So,
.
Answer:
In the figure, m || n and t is a transversal. If
find m5 and m1.
Answer:
Multiple-Choice Test Item
If
and D and E are complementary,
what is mE?
A 53°
C 127°
B 37°
D 7°
Read the Test Item
Since
are complementary,
.
Solve the Test Item
Complementary angles
Replace
with 53°.
Subtract 53 from each side.
Answer: The answer is B.
Multiple-Choice Test Item
If
and G and H are supplementary,
what is mh?
A 76°
B 104°
C 83°
D 14°
Answer: A
Angles PQR and STU are supplementary. If
and
, find the
measure of each angle.
Step 1 Find the value of x.
Supplementary angles
Substitution
Combine like terms.
Add 80 to each side.
Divide each side by 2.
Step 2 Replace x with 130 to find the measure of
each angle.
Answer:
Angles ABC and DEF are complementary. If
and
, find the
measure of each angle.
Answer:
Transportation A road crosses railroad tracks at an
angle as shown. If
find m6 and m5.
Since
are
corresponding angles, they
are congruent.
Answer:
Since
are
supplementary angles, the
sum of their measures is 180°.
180 – 131 = 49
Answer:
Transportation Main Street crosses Broadway
Boulevard and Maple Avenue at an angle as shown.
If m1 = 48°, find m3 and m4.
Answer:
Example 1 Name Corresponding Parts
Example 2 Use Congruence Statements
Example 3 Find Missing Measures
Name the corresponding
parts in the congruent
triangles shown. Then
complete the congruence
statement.
Answer: Corresponding Angles
Corresponding Sides
One congruence statement is
.
HGI  ?
Name the corresponding parts in the congruent
triangles shown. Then complete the congruence
statement.
ABC  ?
Answer:
If
, complete each congruence statement.
Explore You know the congruence statement. You need
to find the corresponding parts.
Plan
Solve
Use the order of the vertices in
to identify the corresponding parts.
Answer:
M corresponds to Q, and N corresponds to P so
N corresponds to P, and O corresponds to R so
M corresponds to Q, and O corresponds to R so
Examine Draw the triangles, using arcs and slash marks
to show the congruent angles and sides.
If
statement.
Answer:
, complete each congruence
Construction A brace is used to support a tabletop.
In the figure,
. What is the measure
of F ?
F and C are corresponding angles. So, they are
congruent. Since
Answer:
.
What is the length of
corresponds to
Since
,
Answer:
?
. So,
and
.
are congruent.
Art In the figure,
.
a. What is the measure of B?
Answer: 44°
b. What is the length of
Answer: 22 in.
?
Example 1 Translation in a Coordinate Plane
Example 2 Reflection in a Coordinate Plane
Example 3 Rotations in a Coordinate Plane
The vertices of ABC are A(–3, 7), B(–1, 0), and
C(5, 5). Graph the triangle and the image of ABC
after a translation 4 units right and 5 units down.
This translation can be written as the ordered pair (4, –5).
To find the coordinates of the translated image, add 4 to
each x-coordinate and add –5 to each y-coordinate.
vertex
translation
4 right, 5 down
A(–3, 7)

(4, –5)

A(1, 2)
B(–1, 0)

(4, –5)

B(3, –5)
C(5, 5)

(4, –5)

C(9, 0)
The coordinates of the vertices of  ABC are A(1, 2),
B(3, –5), and C(9, 0). Graph  ABC and  ABC.
Answer:
The vertices of DEF are D(–1, 5), E(–3, 1), and
F(4, –4). Graph the triangle and the image of DEF
after a translation 3 units left and 2 units up.
Answer:
The vertices of a figure are M(–8, 6), N(5, 9), O(2, 1),
and P(–10, 3). Graph the figure and the image of the
figure after a reflection over the y-axis.
To find the coordinates of the vertices of the image after a
reflection over the y-axis, multiply the x-coordinate by –1
and use the same y-coordinate.
vertex
M(–8, 6) 
N(5, 9) 
O(2, 1) 
P(–10, 3) 




reflection
M(8, 6)
N(–5, 9)
O(–2, 1)
P(10, 3)
The coordinates of the vertices of the reflected figure are
M(8, 6), N(–5, 9), O(–2, 1) and P(10, 3). Graph the
figure and its image.
Answer:
4
–8
–4
4
–4
8
The vertices of a figure are Q(–2, 4), R(–3, 1),
S(3, –2), and T(4, 3). Graph the figure and the image
of the figure after a reflection over the y-axis.
Answer:
A figure has vertices A(–4, 5), B(–2, 4), C(–1, 2),
D(–3, 1), and E(–5, 3). Graph the figure and the image
of the figure after a rotation of 180°.
To rotate the figure, multiply both coordinates of each
point by –1.
A(–4, 5)
B(–2, 4)
C(–1, 2)
D(–3, 1)
E(–5, 3)





A(4, –5)
B(2, –4)
C(1, –2)
D(3, –1)
E(5, –3)
The coordinates of the vertices of the rotated figure are
A(4, –5), B(2, –4), C(1, –2), D(3, –1), and E(5, –3).
Graph the figure and its image.
Answer:
A figure has vertices A(2, –1), B(3, 4), C(–3, 4),
D(–5, –1), and E(1, –4). Graph the figure and the image
of the figure after a rotation of 180°.
Answer:
Example 1 Find Angle Measures
Example 2 Classify Quadrilaterals
Find the value of x. Then find each missing angle
measure.
Words
The sum of the measures of the angles is 360°.
Variable
Let mQ, mR, mS, and mT represent
the measures of the angles.
Equation
Angles of a
quadrilateral
Substitution
Combine like
terms.
Subtract 185
from each side.
Simplify.
Divide each side
by 5.
Answer: The value of x is 35. So,
and
.
Find the value of x. Then find each missing angle
measure.
Answer:
Classify the quadrilateral using the name that best
describes it.
The quadrilateral has one pair of opposite sides parallel.
Answer: It is a trapezoid.
Classify the quadrilateral using the name that best
describes it.
The quadrilateral has both pairs of opposite sides parallel
and congruent.
Answer: It is a parallelogram.
Classify the quadrilateral using the name that best
describes it.
The quadrilateral has four congruent sides and four
right angles.
Answer: It is a square.
Classify each quadrilateral using the name that best
describes it.
a.
b.
Answer: rectangle
c.
Answer: parallelogram
Answer: trapezoid
Example 1 Find Areas of Parallelograms
Example 2 Find Areas of Triangles
Example 3 Find Area of a Trapezoid
Example 4 Use Area to Solve a Problem
Find the area of the parallelogram.
The base is 3 meters.
The height is 3 meters.
Area of a parallelogram
Replace b with 3 and
h with 3.
Multiply.
Answer: The area is 9 square meters.
Find the area of the parallelogram.
The base is 4.3 inches.
The height is 6.2 inches.
Area of a parallelogram
Replace b with 4.3
and h with 6.2.
Multiply.
Answer: The area is 26.66 square inches.
Find the area of each parallelogram.
a.
b.
Answer: 12 cm2
Answer: 1.95 ft2
Find the area of the triangle.
The base is 3 meters.
The height is 4 meters.
Area of a triangle
Replace b with 3
and h with 4.
Multiply.
Multiply.
Answer: The area of the triangle is
6 square meters.
Find the area of the triangle.
The base is 3.9 feet.
The height is 6.4 feet.
Area of a triangle
Replace b with 3.9
and h with 6.4.
Multiply.
Multiply.
Answer: The area of the triangle is 12.48 square feet.
Find the area of each triangle.
a.
b.
Answer:
Answer:
Find the area of the trapezoid.
The height is 6 meters.
The bases are
meters and
meters.
Area of a trapezoid
Replace h with 6 and
a with
and b with
.
Divide out the common
factors.
Simplify.
Answer: The area of the trapezoid is
square meters.
Find the area of the trapezoid.
Answer:
Painting A wall that needs to be painted is 16 feet wide
and 9 feet tall. There is a doorway that is 3 feet by 8 feet
and a window that is 6 feet by
feet. What is the area
to be painted?
To find the area to be painted, subtract the areas of the
door and window from the area of the entire wall.
Area of the wall
Area of the door Area of the window
Answer: The area to be painted is 144 – 24 – 33
or 87 square feet.
Gardening A garden needs to be covered with fresh
soil. The garden is 12 feet wide and 15 feet long. A
rectangular concrete path runs through the middle of
the garden and is 3 feet wide and 15 feet long. Find the
area of the garden which needs to be covered with
fresh soil.
Answer:
Example 1 Classify Polygons
Example 2 Measures of Interior Angles
Example 3 Find Angle Measure of a Regular Polygon
Classify the polygon.
This polygon has 5 sides.
Answer: It is a pentagon.
Classify the polygon.
This polygon has 7 sides.
Answer: It is a heptagon.
Classify each polygon.
a.
b.
Answer: hexagon
Answer: heptagon
Find the sum of the measures of the interior angles
of a quadrilateral.
A quadrilateral has 4 sides. Therefore,
.
Replace n with 4.
Simplify.
Answer: The sum of the measures of the interior angles
of a quadrilateral is 360°.
Find the sum of the measures of the interior angles
of a pentagon.
Answer: 540°
Traffic Signs A stop sign is a regular octagon. What is
the measure of one interior angle in a stop sign?
Step 1 Find the sum of the measures of the angles. An
octagon has 8 sides. Therefore,
.
Replace n with 8.
Simplify.
The sum of the measures of the interior angles is 1080°.
Step 2 Divide the sum by 8 to find the measure
of one angle.
Answer: So, the measure of one interior angle
in a stop sign is 135°.
Picnic Table A picnic table in the park is a regular
hexagon. What is the measure of one interior angle
in the picnic table?
Answer: 120°
Example 1 Find the Circumference of a Circle
Example 2 Use Circumference to Solve a Problem
Example 3 Find Areas of Circles
Find the circumference of the circle to the
nearest tenth.
Circumference of a circle
Replace d with 12.
Simplify. This is the
exact circumference.
To estimate the circumference, use a calculator.
12
2nd
[]
ENTER
37.69911184
Answer: The circumference is about 37.7 meters.
Find the circumference of the circle to the
nearest tenth.
Circumference of a circle
Replace r with 7.1.
Simplify. Use a calculator.
Answer: The circumference is about 44.6 meters.
Find the circumference of each circle to the
nearest tenth.
a.
b.
Answer: 12.6 ft
Answer: 10.1 cm
Landscaping A landscaper has a tree whose roots
form a ball-shaped bulb with a circumference of
about 110 inches. How wide will the landscaper have
to dig the hole in order to plant the tree?
Explore You know the circumference of the roots of the
tree. You need to know the diameter of the hole
to be dug.
Plan
Use the formula for the circumference of a circle
to find the diameter.
Circumference of a circle
Solve
Replace C with 110.
Divide each side by .
Simplify. Use a calculator.
Answer: The diameter of the hole should be at
least 35 inches.
Examine Check the reasonableness of the solution by
replacing d with 35 in
.
Circumference of a circle
Replace d with 35.
Simplify. Use a calculator.
The solution is reasonable.
Swimming Pool A circular swimming pool has a
circumference of 24 feet. Matt must swim across the
diameter of the pool. How far will Matt swim?
Answer: about 7.6 ft
Find the area of the circle. Round to the nearest tenth.
Area of a circle
Replace r with 11.
Evaluate
.
Use a calculator.
Answer: The area is about 380.1 square feet.
Find the area of the circle. Round to the nearest tenth.
Area of a circle
Replace r with 4.15.
Evaluate
.
Use a calculator.
Answer: The area is about 54.1 square centimeters.
Find the area of each circle. Round to the nearest tenth.
a.
b.
Answer:
Answer:
Example 1 Find Area of Irregular Figures
Example 2 Use Area of Irregular Figures
Find the area of the figure to the nearest tenth.
Explore You know the dimensions
of the figure. You need to
find its area.
Plan
Solve a simpler problem.
First, separate the figure
into a triangle, square,
and a quarter-circle. Then
find the sum of the areas
of the figure.
Find the area of the figure to the nearest tenth.
Solve
Area of Triangle
Area of a triangle
Replace b with 2
and h with 4.
Simplify.
Find the area of the figure to the nearest tenth.
Solve
Area of Square
Area of a square
Replace b and h with 2.
Simplify.
Find the area of the figure to the nearest tenth.
Solve
Area of Quarter-circle
Area of a quarter-circle
Replace r with 2.
Simplify.
Answer: The area of the figure is
or about 11.1 square inches.
Find the area of the figure to the nearest tenth.
Answer:
Carpeting Carpeting costs $2 per square foot. How
much will it cost to carpet the area shown?
Step 1 Find the area to be carpeted.
Area of Rectangle
Area of a rectangle
Replace b with 14
and h with 10.
Simplify.
Carpeting Carpeting costs $2 per square foot. How
much will it cost to carpet the area shown?
Area of Square
Area of a square
Replace b and h with 3.
Simplify.
Carpeting Carpeting costs $2 per square foot. How
much will it cost to carpet the area shown?
Area of Triangle
Area of a triangle
Replace b with 14
and h with 12.
Simplify.
The area to be carpeted is
or 233 square feet.
Carpeting Carpeting costs $2 per square foot. How
much will it cost to carpet the area shown?
Step 2 Find the cost of the carpeting.
Answer: So, it will cost $466 to carpet
the area shown.
Painting One gallon of paint is advertised to cover
100 square feet of wall surface. About how many
gallons will be needed to paint the wall shown below?
Answer: about 4 gallons