Geometry First Semester Practice Final (cont)
49.
Determine the width of the river, AE, if
A.
6.6 yards
B.
10 yards
C.
12.8 yards
D.
15 yards
K
50.
M
In the similar triangles shown below, what is
the value of x?
A.
13.5
18
B.
16.7
C.
18
D.
24
J
20
15
L
x
N
51.
. Determine the length of the longest side of
A.
B.
C.
D.
222
576
648
666
A
576
192
B
216
Which additional piece of information would prove that
N
A.
NM = 18
B.
LM = 18
C.
NM = 15
12
D.
LM = 10
L
For
,
Find the length of
A.
3
B.
12
C.
10
D.
20
6
.
Q
14
X
Geometry
R
7
Z
(cont)
101
Z
Y
?
K
12
8
I
M
Y
53.
X
222
C
52.
.
10
J
Geometry First Semester Practice Final (cont)
54.
Lines l, m, and n are parallel. Determine the value of x.
A.
2.7
4
B.
6.0
C.
8.5
D.
9.4
x
l
5
m
7. 5
n
55.
. Determine the value of x.
A.
B.
C.
D.
35
5
20
30
N
20
B
C
16
Determine the length of
A.
19
B.
39
C.
52
D.
14
25
12
M
A
56.
15
L
x
.
B
6
D
(2x + 1)
8
A
Geometry
102
(3x – 5)
C
Unit 7 Objective 0
Determine the value of the missing angle in each of the figures below:
NOTE: DIAGRAMS ARE NOT DRAWN TO SCALE.
1.
2.
3.
4.
5.
6.
7.
8.
(cont)
Geometry
103
Unit 7 Objective 0 (cont)
Determine the value of the missing angle in each of the figures below:
NOTE: DIAGRAMS ARE NOT DRAWN TO SCALE.
9.
10.
11.
12.
13.
14.
Geometry
104
Geometry Unit 7
1.
Apply the Pythagorean Theorem (Section 7.1)
2.
Use the converse of the Pythagorean Theorem (Section 7.2)
3.
Use Similar Right Triangles (Section 7.3)
4.
Special Right Triangles (Section 7.4)
5.
Apply the Tangent Ratio (Section 7.5)
6.
Apply the Sine and Cosine Ratios (Section 7.6)
7.
Solve Right Triangles (Section 7.7)
Review
Geometry
105
Unit 7 Trig Worksheet 1
SOH CAH TOA
Remember
sin =
tan =
cos =
C
Example 1:
Find the value of
Solution:
cos A •
=
=
cos A • tan B
5
4
B
3
tan C
•
•
=
B
Example 2:
Find the value of
Solution:
sin 2 A =
sin 2 A
(sin A)
13
12
•
=
•
=
•
(sin A)
5
C
=
N
C
Homework
Use
shown to find the values in problems 1-6
1. sin L • cos L
2. tan N • sin L
4. tan 2 L
5. 1 + cos 2 L
(cont)
Geometry
A
106
10
L
8
3. cos L •2tan N
0
2
6. sin L f+ cos 2 L
e
e
t
h
h
6
M
B
Unit 7 Trig Worksheet 1 (cont)
Use
B
the triangle shown to find the values in problems 7-15.
7.
sin A • cos B
8.
tan B • sin A
9.
sin 2 A – 1
10.
1 – cos 2 A
11.
tan 2 B + 1
12.
13.
14.
15.
Geometry
107
5
3
C
4
A
Unit 7 Trig Worksheet 2
Notes:
Let’s review the formulas for sin, cos, tan.
sin =
cos =
tan =
We have been finding these values using triangles with numbers.
C
Example 1 Find sin A
5
Solution: sin A =
4
=
A
B
3
2
In this lesson we are not going to have numbers on the triangle.
0
R
f
Example 2
Find sin K and tan R
e
e
t
h
Solution:
K
P
sin K =
(just puththe letters of the sides) =
tan R =
=
B
Example 3
Determine if the statement is true or false using the
given triangle.
C
Solution:
A
AB • cos B = BC (get the trig word by itself by dividing both sides by AB)
•cos B =
cos B =
=
which is the cos formula, so True
(cont)
Geometry
108
Unit 7 Trig Worksheet 2 (cont)
Homework:
Using the given triangle find the requested values in problems 1 -4 (Just put the letters of the
sides.)
U
V
W
1.
tan W
2.
sin U
3.
Draw your own right triangle and label it ∆JKL.
Use this triangle to find the values in problems 5-7.
(Just put the letters of the sides.)
5.
sin J
6.
tan L
7.
cos U
4.
tan U
is a right angle.
cos J
Use the given triangle to determine whether the statements in problems 8-15 are true or false.
B
8.
sin A =
9.
cos B =
10.
tan A =
11.
cos A =
C
A
(In Problems 12 -13 remember to get the trig word by itself by dividing.)
12.
BA • sin A = BC
13.
BC • tan B = AC
(In problems 14-15, put all the letters of the sides and compare the left side of the equation with
the right side of the equation.)
14.
sin A = cos B
Geometry
15.
sin A = tan A • cos A
109
Unit 7 Trig Worksheet 3
Notes:
Determine whether each equation below is true or false:
1. cos 2 A + 1 = sin 2 A
2. sin 2 A + cos 2 A = 1
Solution:
a) First make a right triangle .
b)
Put any right triangle Pythagorean numbers on your triangle. You could use 3,4,5 or
5,12,13. Let’s use 3,4,5
(Remember the largest number is the hypotenuse.
Also label one of the non-right angles as A
c) Now, find sin A
5
4
=
A
3
Find cos A
=
d)
Now plug the fractions into each problem and see if it’s true or false.
1.
cos 2 A
sin 2 A
2.
cos 2 A =
=
+
1
=
+
= 1
+
1
=
+
=
1
=
1
=
This is not a true equation, so the
original trig problem was false.
1
This is a true statements, so the
original trig problem was true.
(cont)
Geometry
sin 2 A +
+ 1
110
Unit 7 Trig Worksheet 3 (cont)
Homework:
Use the same triangle that we used in our note examples and give the following fraction values.
1. sin A = ________
2. cos A = ________
3. tan A = ________
Now, using the above fraction values, determine whether each of the following is true or false.
4. sin A • cos A = tan A
5. tan A • cos A = sin A
6. tan A • sin A = cos A
7. 1 – sin 2 A = cos 2 A
8. cos 2 A – sin 2 A = 1
9. cos A =
10. 1 + tan 2 A = cos 2 A
11. tan A =
12. sin 2 A = tan 2 A • cos 2 A
Unit 7 Review
NOTE: Diagrams are not drawn to scale.
Select the correct multiple choice response:
1.
The lengths of the two legs of a right triangle are 4 and 7. What is the length of the
hypotenuse?
A.
33
B.
C.
65
D.
2.
The lengths of the two legs of a right triangle are
and 5. What is the length of the
hypotenuse?
A.
B.
C.
D.
(cont)
Geometry
111
Unit 7 Review (cont)
3.
The length of one leg of a right triangle is
Find the length of the other leg.
A.
25
B.
5
C.
D.
4.
A baseball diamond is in the shape of a square,
90 feet on each side. What is the direct distance
from home plate to second base?
A.
90 ft
B.
ft
C.
ft
D.
180 ft
5.
A model rocket is launched. It rises to a point 36 feet
above the ground and is 48 feet along the ground from the
lift off site, as shown in the diagram.
What is the length of the rocket’s path in the air?
A.
12 ft.
B.
32 ft.
C.
60 ft.
x
D.
84 ft.
Rocket
Launch
Site
and the hypotenuse is 11.
Current
Site of
Rocket
36 ft
48 ft
6.
Which set of numbers can represent the side lengths of an obtuse triangle?
A.
5, 10, 11
B.
3, 5,
C.
3, 7, 8
D.
1, 2, 2
7.
Which set of numbers below represent the lengths of the sides of a right triangle?
A.
1, 2,
B.
5, 11, 12
C.
6, 8,
D.
5, 7, 9
(cont)
Geometry
112
Unit 7 Review (cont)
A
8.
9.
10.
11.
12.
13.
Determine the length of BD .
A.
13
B.
C.
36
D.
6
Determine the value of x.
A.
B.
C.
D.
9
D
4
B
C
x
6
1
12
Determine the value of x.
A.
B.
C.
13
D.
Determine the value of x
A.
B.
6
C.
3
D.
5
x
x
45°
Determine the height of the triangle if AB = 16 cm and BC = 17 cm
A.
cm
B.
cm
C.
D.
9 cm
15 cm
Determine the values of x and y.
A.
x=
,y=3
B.
x=
,y=
C.
x = 3, y =
D.
x=
,y=
h
A
y
x
60°
6
(cont)
Geometry
C
113
B
Unit 7 Review (cont)
14.
Determine the value of x.
A.
40
B.
60°
C.
D.
15.
16.
17.
x
Which equation is equivalent to
A.
x=
B.
x=
C.
D.
x = 17 cos A
x = cos (17A)
cos A =
Determine the value of x to the nearest tenth.
A.
3.3
B.
2.7
C.
7.1
D.
6.4
?
3
25°
x
In right triangle DEF, DE = 15, EF = 36 and DF = 39.
What is the cos F?
A.
B.
C.
D.
18.
An 80 foot support wire is attached to the top of a tower and
meets the ground at a 70° angle.
How tall is the tower, to the nearest foot?
A.
27 ft.
sin 70° = 0.94
sin 20° = 0.34
B.
70 ft.
cos
70°
=
0.34
cos 20° = 0.94
C.
220 ft.
tan
70°
=
2.75
tan 20° = 0.36
D.
75 ft.
(cont)
Geometry
114
Unit 7 Review (cont)
19.
A ladder is leaning against a tree and hits the tree at a point
15 feet above the ground. The ladder and the ground
form a 62° angle. How far, to the nearest tenth of a foot,
is the bottom of the ladder from the base of the tree?
A.
13.2 ft.
B.
7 ft.
sin 62° = 0.88
sin 28° = 0.47
C.
28.2 ft.
cos 62° = 0.47
cos 28° = 0.88
D.
8 ft.
tan 62° = 1.88
tan 28° = 0.53
20.
Which expression is correct?
A.
cos
=
B.
cos
=
f
C.
cos
=
D.
cos
=
P
d
e
21.
The angle of depression from the top of a 20-foot
lighthouse to a boat in the ocean is 39°. Which is closest
to the distance that the boat is from the base of the lighthouse?
A.
12.6 ft
B.
16.2 ft
sin 39° = 0.63
sin 51° = 0.78
C.
24.6 ft
cos 39° = 0.78
cos 51° = 0.63
D.
15.6 ft
tan 39° = 0.81
tan 51° = 1.23
22.
The angle of elevation from the boy to the top of the
flagpole is 35°. How far (to the nearest tenth)
is the boy from the base of the flagpole?
A.
85.7 ft.
B.
42 ft.
C.
49 ft.
D.
34.4 ft
sin 35° = 0.57
cos 35° = 0.82
tan 35° = 0.70
Geometry
sin 55° = 0.82
cos 55° = 0.57
tan 55° = 1.43
115
Unit 8 Objective 0
1.
If possible, draw 2 obtuse triangles that are similar. Label their measurements and state
whether they are similar by AA, SSS, or SAS.
If not possible, state that it can’t be done.
2.
If possible, draw 2 obtuse triangles that are not similar. Label their measurements and
state why they are not similar.
If possible, draw 2 scalene triangles with congruent bases that are similar.
Label their measurements and state whether they are similar by AA, SSS, or SAS.
If not possible, state that it can’t be done.
3.
4.
If possible, draw 2 scalene triangles with congruent bases that are not similar. Label their
measurements and state why they are not similar.
5.
If possible, draw 2 right triangles that are similar.
Label their measurements and state whether they are similar by AA, SSS, or SAS.
If not possible, state that it can’t be done.
6.
If possible, draw 2 right triangles that are not similar. Label their
measurements and state why they are not similar.
7.
If possible, draw 2 isosceles triangles with congruent vertex angles that are similar.
Label their measurements and state whether they are similar by AA, SSS, or SAS.
If not possible, state that it can’t be done.
8.
If possible, draw isosceles triangles with congruent vertex angles that are not similar.
Label their measurements and state why they are not similar.
9.
In the figure below, determine if the value of n could be 1.
n
Explain why or why not?
n
8
10.
In the figure below, determine if the value of n could be 4.
Explain why or why not?
n
n
8
11.
In the figure below, determine if the value of n could be 5.
Explain why or why not?
n
n
8
12.
The complement of an angle is four times the measure of the angle itself. Calculate the
angle and its complement.
(cont)
Geometry
116
Unit 8 Objective 0 (cont)
13.
The measures of the interior angles of a triangle are (6x + 6)°, (4x – 4)°, and (4x + 10)°.
Calculate the degree measures of all three angles.
14.
The four sides of the figure will be folded up and taped
to make an open box. What will be the volume of the
box?
15.
Calculate the value of x.
16.
Calculate the value of x.
Y
17.
Supply the reasons in the proof below:
Given:
Prove:
1.
;
W
bisects
X
Statements
;
bisects
Reasons
1.
2.
2.
3.
3.
4.
4.
Geometry
117
Z
Geometry Unit 8
1.
Find angle measure in polygons. Interior angle sums, exterior angle sums, etc.
(Section 8.1)
2.
Use properties of parallelograms. (Section 8.2)
3.
Show that a quadrilateral is a parallelogram. (Section 8.3)
4.
Properties of rhombuses, rectangles, and squares (Section 8.4)
5.
Use properties of trapezoids and kites. (Section 8.5)
6.
Identify special quadrilaterals (Section 8.6)
Review
Geometry
118
Unit 8 Worksheet 2
Determine the values for the variables that make the quadrilateral a parallelogram.
1.
2.
3.
m + 2n
11
15
2x + 4y
16
3m + n
x + 3y
5
a–b
7
a+b
9
Determine the coordinates of the point where the diagonals of the parallelogram intersect.
4.
5.
y
y
(5,7)
(12,7)
(3,4)
(m + w, n)
(m, n)
(10,4)
x
(0, 0)
(w, 0)
x
Worksheet 3
In parallelogram ABCD, diagonals
and
intersect at point V.
By theorem 8.10 we know that diagonals bisect each other so BV = DV and AV = CV as shown.
We also know that opposite sides are congruent in a parallelogram.
Use this information and the diagram to answer true or false to the
statements below:
1.
A
B
2.
3.
V
4.
D
Geometry
119
C
Unit 8 Practicing Proofs
1.
Given: Parallelogram PQRS;
S
K
R
Prove:
P
J
Statements
1.
PQRS;
Q
Reasons
1.
2.
2.
3.
3.
4.
4.
5.
5.
T
2.
3
Given:
;
Prove: QRST is a parallelogram
1
4
Q
Statements
1.
;
Reasons
1.
2.
2.
3.
3.
4.
4.
5.
5.
6. QRST is a parallelogram .
6.
(cont)
Geometry
R
120
S
2
Unit 8 Practicing Proofs (cont)
D
3. Given:
Prove:
C
y°
x°
is a parallelogram.
x°
y°
B
A
Statements
1.
;
Reasons
1.
2.
2.
3.
3.
4.
4.
5.
is a parallelogram
5.
.
Geometry
121
Unit 8 Worksheet 4
1.
Look at the coordinate grid below. Two points are to be added to the grid to form a
square.
a. Place two points in Quadrant 2
b. Place two points in Quadrant 3
that would form a square with
that would form a square with
the existing points.
the existing points.
Give the coordinates of the two points
Give the coordinates of the two points.
Complete the sketch of the square.
Complete the sketch of the square.
2.
Look at the coordinate grid below. Two points are to be added to the grid to form a
rectangle with an area of 20 square units.
Place two points in Quadrant 4 so
b. Place two points in Quadrant 1 so
that a rectangle with an area of
that a rectangle with an area of
20 sq. units is formed.
20 sq. units is formed.
Give the coordinates of the two points.
Give the coordinates of the two
Complete the sketch of the rectangle.
points.
Complete the sketch of the
rectangle.
a.
3.
Look at the coordinate grid below. Point D is to be added in Quadrant 1to form a square.
The slope of
a.
b.
c.
is
and the slope of
.
Use the slope information to help you plot point D to form a square. Complete
the sketch of the square. Give the coordinates of point D.
Use slopes to show that ABCD has 4 right angles
A
Use the distance formula to show that all
4 sides in ABCD are congruent.
B
(cont)
Geometry
is
122
C
Unit 8 Worksheet 4 (cont)
4.
The points (2, 1), (5, 1) and (2, – 1) are plotted below.
a. Plot a fourth point in quadrant 4 that will
make a parallelogram.
Give the coordinate of that point.
Complete the sketch of the parallelogram.
b. Plot a fourth point in quadrant 1 that will
make a parallelogram.
Give the coordinate of that point.
Complete the sketch of the parallelogram.
c.
Plot a fourth point in quadrant 3 that will
make a parallelogram.
Give the coordinate of that point.
Complete the sketch of the parallelogram.
5.
The points (–1, 1), (–1, 3) and (1, 1) are plotted below.
a. Plot a fourth point in quadrant 1 that will
make a parallelogram.
Give the coordinate of that point.
Complete the sketch of the parallelogram.
b. Plot a fourth point in quadrant 2 that will
make a parallelogram.
Give the coordinate of that point.
Complete the sketch of the parallelogram.
c. Plot a fourth point in quadrant 4 that will
make a parallelogram.
Give the coordinate of that point.
Complete the sketch of the parallelogram.
Geometry
123
Unit 8 Review
In problems 1 – 22, choose the correct multiple choice response.
NOTE: Diagrams are not drawn to scale.
1.
What is the value of x?
120°
A.
540°
B.
390°
C.
150°
x°
D.
120°
2.
Determine the value of x.
A.
15
B.
15.4
C.
9
D.
19.8
(8x + 1)°
95°
73°
(5x – 4)°
3.
Determine the sum of the exterior angles of an octagon.
A.
1440°
B.
1080°
C.
360°
D.
135°
4.
Determine the measure of each interior angle of a regular sided polygon with 9 sides.
A.
1620°
B.
180°
C.
1260°
D.
140°
5.
Determine the measure of each exterior angle of a regular polygon with 12 sides.
A.
30°
B.
150°
C.
216°
D.
36°
6.
The measure of an interior angle of a regular polygon is 162°. How many sides does the
polygon have?
A.
18 sides
B.
20 sides
C.
16 sides
D.
10 sides
7.
Determine the value of x?
A.
80°
B.
40°
C.
60°
D.
20°
100°
140°
100°
100°
140°
(cont)
Geometry
124
x°
Unit 8 Review (cont)
8.
If the sum of the interior angles of a polygon equals 3780°, how many sides
does the polygon have?
A.
23
B.
21
C.
20
D.
19
9.
What is the measure of each interior angle of a regular hexagon?
A.
60°
B.
135°
C.
45°
D.
120°
10.
What are the values of the variables in the
given parallelogram?
A.
x=7,y=9
B.
x = 7 , y = 65
(6x – 8)°
C.
x = 5 , y = 71
D.
x = 3 , y = 77
G
F
If FH = 30, find FK.
A.
12
K
B.
18
C.
15
12
D.
30
11.
13.
14.
PQVT
A.
B.
C.
D.
(2y + 16)°
H
J
12.
(4x + 6)°
is a rhombus. Determine the value of x.
110°
55°
70°
35°
T
If PQRS is a rhombus, which statement must be true?
A.
is a right angle
B.
C.
P
D.
Which statement is true?
A.
All quadrilaterals are rectangles
B.
All rectangles are quadrilaterals
C.
All rectangles are squares
D.
All quadrilaterals are squares
(cont)
Geometry
125
P
110°
Q
x°
V
S
R
Q
Unit 8 Review (cont)
15.
16.
17.
Determine the value of x.
A.
4
B.
6
C.
12
D.
8
2y – 10
x+6
3x – 6
y+2
In the diagram
1 = 9x,
2=x+y
Determine the values of x and y
A.
x = 20°, y = 165°
B.
x = 10°, y = 80°
C.
x = 20°, y = 160°
D.
x = 10°, y = 85°
1
2
and
are congruent base angles of isosceles trapezoid JKLM.
If
= (18x + 5)°,
= (14x + 15)° and
= (17x + 10)°, determine
the value of x.
A.
18.
19.
20.
B.
2
C.
15
D.
5
The perimeter of square MNOP is 72 inches, and NO = 2x + 6. What is the value of x?
A.
15
B.
12
C.
6
D.
9
K
L
4x + 1
Determine the length of
in the trapezoid shown.
A.
26
5x + 2
H
I
B.
4
27
C.
13
M
N
D.
17
Which quadrilateral has two pairs of consecutive congruent sides, but opposite sides are
not congruent?
A.
Kite
B.
Rhombus
C.
Trapezoid
D.
Parallelogram
(cont)
Geometry
126
Unit 8 Review (cont)
21.
22.
What is the most specific name for the figure shown?
A.
Quadrilateral
B.
Parallelogram
C.
Trapezoid
D.
Rectangle
91°
2x + y
Determine the value of x in the given parallelogram.
A.
2
B.
3
4x + 3y
C.
4
D.
6
24
10
2x + 7y
Work the following, showing all work.
23.
7
Determine the values of x and y.
11
3x + y
24.
ABCD is an isosceles trapezoid with midsegment
10
B
n=
C
(3x + 2)°
___________
EF = ___________
3n – 4
E
. Determine the following:
x=
F
___________
= _______
(2x – 17)°
A
25.
36
D
ABCD is a parallelogram. Determine the following.
B
3x + 2y
E
7x – y
C
z+6
21
z2
A
26
___________
y=
___________
z=
___________
AC = ___________
D
(cont)
Geometry
x=
127
Unit 8 Review (cont)
26.
ABCD is a rectangle with perimeter 96 meters. Determine the following.
B
A
= ___________
28°
E
2n + 8
= ___________
n=
D
27.
6n – 12
___________
BC = ___________
C
Given a 25-gon:
a.
What is the sum of the measures of the interior angles?
b.
What is the sum of the measures of the exterior angles?
28.
Sketch LMNP if L (2, 1), M (1, 4), N (7, 6), and P (8, 3). Prove that LMNP is a
rectangle.
29.
The points (1, –1), (4, –1) and (1, – 4) are plotted below.
A. Plot a fourth point in
quadrant 4 that will make
a parallelogram.
Geometry
B. Plot a fourth point in
quadrant 3 that will make
a parallelogram.
128
C. Plot a fourth point in
quadrant 1 that will
make a parallelogram.
Unit 8 Systems Practice
What values must x and y have to make each quadrilateral a parallelogram?
1.
2.
(8x − 6)o
3y
o
y2
(3x − 40)
x
( y + 30)
42o
3.
4.
42
(7y − 2)o
(4x + 1)o
3x − 2y
26
9yo
4x + y
Geometry
129
3xo
Unit 8 Quadrilateral & Polygon Worksheet
[1-12]: Solve for the variables. Give the best name for each of the following based upon
given information and calculations. Show logical & appropriate work!
{Names of Quadrilaterals are: Quadrilateral, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, & Isosceles
Trapezoid}
1.
5x o
2.
4y o
1. x = __________
y = __________
Name:____________
xo
y
70 o
2. x = __________
o
y = __________
3.
x
4.
o
y
2n + 40
Name___________
115 o
3. x = __________
o
37
y = __________
o
(12x
19) o
Name:____________
4n + 12
4. x = __________
y = __________
5.
6.
(8n
Name:____________
5. x = __________
Perimeter of quadrilateral
is 274 meters.
11)
y = __________
n = __________
xo
(5n + 1)
y
o
52
Name:____________
3x +1
o
6. x = __________
8x
7.
7
8.
Name ___________
7.
Perimeter of quadrilateral is 90 cm
(7x
o
(x
4) feet
(5x + 3) o
15 feet
3n + 2
A
24 feet
5n + 7
Name____________
8.
D
x = __________
BC = ________
Name___________
(cont)
Geometry
y = _________
C
n = _________
yo
24 o
B
19)
x = _________
130
Unit 8 Quadrilateral & Polygon Worksheet (cont)
9.
3x + 5
B
10.
C
2x
B
46
5x
A
1
D
A
3x
(4a + 1)o
3
24
63o
Name = _________
10. x = __________
(6a
13)o
z = __________
D
A
x
(2y + 1)o
o
5m
6
11. x = _________
B
D
y = _________
C
xo
7n + 3
17
Name ___________
Quadrilateral ABCD is a rhombus.
C
A
yo
25 o
x = __________
AD = __________
12.
Quadrilateral ABCD is a parallelogram.
13.
9.
C
2x + 1
11.
B
2
m = _________
n = _________
12. x = _________
y = _________
z
o
z = _________
D
=_____
Find the sum of the measures of the interior angles for the following convex polygons.
a. 17-gon
b. 34-gon
c. 51-gon
13.a._____________
b.____________
c.____________
14.
Find the measure of each exterior angle for the following regular polygons.
a. Pentagon
b. Heptagon
c. 45-gon
14.a._____________
b.____________
c.____________
15.
Find the measure of each interior angle for the following regular polygons. 15.a.______________
a. Decagon
b. Octagon
c. 21-gon
b.____________
c.___________
16.
Find the number of sides for a convex polygon whose interior angle sum is: 16a.________
a. 3060o
b. 5400o
c. 4500o
b.________
c.________
17.
Find the number of sides for the following regular polygons, given:
a. The measure of each exterior angle is 7.5o.
b. The measure of each interior angles is 157.5°
Geometry
131
17.a.___________
b.___________
Quadrilateral Tree
Quadrilateral
Kite
1.
Trapezoid
1.
2.
1.
3.
Parallelogram
Isosceles
Trapezoid
1.
2.
1.
3.
2.
4.
5.
Rectangle
3.
Rhombus
1.
1.
2.
2.
3.
Please note: Midsegment = average of the
bases
Square
Base # 2
midsegment
1.
Base # 1
or
Geometry
132
Unit 8 Extra Practice
In problems 1-3 find the requested parts.
(Remember names of Quadrilaterals are: Quadrilateral, Parallelogram, Rectangle, Rhombus,
Square, Trapezoid, Isosceles Trapezoid, & Kite}
1.
Perimeter = 144 meters
x = __________
yo
y = __________
n = __________
3n +13
xo
Best Name:____________
52 o
.
8n − 7
2.
x = __________
2n − 4
B
C
= __________
n = __________
2n + 5
= __________
A
(5x + 3) o
(3x + 35) o
4n + 5
Best Name:____________
D
3.
x = __________
yo
(10n − 13)
x
(7n + 5)
o
= __________
n = __________
= __________
65
o
Best Name:____________
4.
Find the measure of each interior angle for a regular 15-gon.
5.
Find the measure of each exterior angle for a regular heptagon.
6.
Find the number of sides for a regular polygon with each exterior angle measuring 15o.
7.
Find the number of sides for a regular polygon with each interior angle measuring 172o.
Geometry
133
Properties of Quadrilaterals
Property
Two pairs of opposite
sides are parallel.
Parallelogram Rectangle Rhombus
Has exactly one pair
of parallel sides.
Two pairs of opposite
sides are congruent.
Has two pairs of
consecutive
congruent sides, but
opposite sides are not
congruent.
All sides are
congruent.
Diagonals are
congruent.
Diagonals are
perpendicular.
A diagonal bisects
two angles.
A diagonal forms two
congruent triangles.
Diagonals bisect each
other.
Opposite angles are
congruent.
All angles are right
angles.
Consecutive interior
angles are
supplementary.
Geometry
134
Square
Trapezoid
Kite
Unit 10 Objective 0
1.
A rhombus has a diagonal that lies on the line y =
x + 1. What is the slope of the other
diagonal of the rhombus?
2.
Find the measure of
in parallelogram ABCD.
Z
A
B
40°
3.
LMNP is a rectangle.
2 and
What is the measure of 1?
3 are congruent. D
L
C
M
1
50°
2
P
4.
In the diagram shown the measure of
What is the measure of 2?
3
N
1 is 4 times as large as the measure of
2
1
5.
Which multiple choice serves as a counterexample to the statement:
All quadrilaterals have four right angles.
A.
A square
B.
A rhombus
C.
A rectangle
6.
In rectangle ABCD, AB =
, CD =
. determine the value of x.
P
7.
PQRS is a rhombus. What is the value of y?
(cont)
x+5
6
S
y
Geometry
135
S
Q
2x
2.
Unit 10 Objective 0 (cont)
8.
Currently the only way to travel from the city of Rio
to the city of Phillie is by traveling through the city of Duke
as shown in the drawing at the right.
Engineers are thinking about building a road directly
from the city of Rio to the city of Phillie as shown below.
a. Determine the number of
miles that the direct route
frm Rio to Phillie would be.
b. Determine how many miles
would be saved by a person
driving the direct route as
compared to a person driving
the long way through Duke.
Original
Route
Phillie
12 miles
New Proposed
Route
5 miles
Duke
Phillie
12 miles
Duke
9.
5 miles
Rio
Given the parallelogram below, find the coordinates for P, without using any new
variables.
y
P
(a, b)
x
(c, 0)
10.
For the quadrilateral shown, find
V
112°
W
Y
47°
Z
Geometry
136
Rio
Geometry Unit 10
1.
Parts of a circle, including tangent lines. (Section 10.1)
2.
Central angles and finding arc measures. (Section 10.2)
3.
Apply properties of chords. (Section 10.3)
4.
Use inscribed angles and inscribed polygons. (Section 10.4)
5.
Interior and exterior angle relationships with circles. (Section 10.5)
6.
Find segment lengths in circles. (Section 10.6)
7.
Write and graph equations of circles. (Section 10.7)
Review
Unit 10 Worksheet 4 Basic Terms & Tangents
In problems 1 – 6 refer to O. Name each of the following:
1.
Two radii ____ and _____ 2.
A diameter ________
3.
A secant _______
4.
A tangent _________
5.
Two chords _____and ___
6.
A point of tangency _____
In problems 7 – 11 refer to B with radius BP. Find the following:
7.
If BP = 4, then SP = _______
8.
If SP = 16n, then BP = ______
9.
If
10.
If
and
are tangent to
then ______
________.
If
is tangent to B, then
B,
would be _____________ to
.
11.
is tangent to
B, then
= ______.
(cont)
Geometry
137
Unit 10 Worksheet 4 Basic Terms & Tangents (cont)
In problems 12 – 14 refer to O.
12.
If
= 60°, then AB = _______
13.
If
= 90°, then BC = _______
14.
Name an inscribed polygon in the figure __________
In problems 15 – 17, O and P are the centers of the circles.
In problem 16,
and
are tangent to both circles and
whose lengths are shown.
15.
16.
OP = ______
RS = ________
In the diagram for Problems 18 – 20,
into segments
17.
HI = _______
is tangent to
18.
If DE = 12 and DO = 9, then OE = ______
19.
If
20.
If DO = 5 and CE = 8, then DE = _____
Geometry
divides
O.
= 60° and OD = 9, then OE = _______
138
Unit 10 Worksheet 5A
In problems 1 – 4,
and
1.
If
= 85° and
are chords.
= 73°, then
= 136° and
1 = ______.
2.
If
= 96°, then
1 = ______.
3.
If
1 = 54° and
= 78°, then
= _______.
4.
If
1 = 48° and
= 42°, then
= _______.
In problems 5 – 7,
and
are tangents.
5.
If
= 280°, then
= ______.
6.
If
= 96°, then
7.
If
= 90°, then
= ______.
= ______.
In problems 8 – 10,
is a tangent.
8.
If
= 120° and
= 40°, then
9.
If
= 45° and
= 55°, then
10.
If
= 50° and
= 110°, then
= ______.
= _______.
= _______.
In problems 11 – 15,
and
11.
If
= 100° and
are secants.
= 20°, then
= _____.
12.
If
= 130° and
= 40°, then
= _____.
13.
If
= 25° and
= 25°, then
14.
If
= 40° and
= 130°, then
15.
If
= 90°,
= 60°, and
= _____.
= _____.
= 80°, then
In problems 16 – 19,
is tangent to the circle at point E.
16.
If
= 100° and
= 20°, then
= _____.
17.
If
= 25° and
= 25°, then
= ______.
18.
If
= 95° and
= 25°, then
= ______.
19.
If
= 40° and
= 138°, then
Geometry
= _____.
139
= _____.
Unit 10 Worksheet 5B
In problems 1 – 6, find the values of a, b, and c.
1.
2.
a = _________
a = _________
b = _________
b = _________
c = _________
c = _________
3.
4.
a = _________
a = _________
b = _________
b = _________
c = _________
c = _________
5.
6.
is a diameter of
a = _________
a = _________
b = _________
b = _________
c = _________
c = _________
O.
is tangent to
O at A.
= 50°.
1 = _____
2 = _____
3 = _____
4 = _____
5 = _____
6 = _____
7 = _____
8 = _____
9 = _____
10 = _____
Geometry
140
= 80°,
= 20°, and
Unit 10 Worksheet 6A
Solve for x. Show your work.
1.
2.
3.
4.
5.
6.
x°
7.
8.
9.
11.
10.
12.
13.
(cont)
Geometry
141
Unit 10 Worksheet 6A (cont)
14.
15.
16.
17.
18.
19.
20.
21. AB = 48
Find the radius.
Given: Circle O
= 124°,
= 140°,
Find the measure of
1 through 10
Geometry
142
= 62°.
Unit 10 Worksheet 6B
In
1.
O,
= 50° and
= 70°. Find each of the following:
2.
3.
4.
5.
6.
In
O,
and
7.
8.
= 60°,
= 80°
= 110°,
is tangent to
O at C,
= 130°. Find each of the following:
1
2
9.
10.
11.
3
12.
O has arc measures as shown.
Find the following:
13.
14.
1
15.
2
16.
3
and
are tangent at J and M, respectively.
(cont)
Geometry
143
Unit 10 Worksheet 6B
17.
is tangent to
O at Q. If
= 15 and PO = 17, find the radius of the circle.
18.
Find the total number of common tangents that can be drawn to two coplanar circles that
are externally tangent.
19.
Complete: If a line in the plane of a circle is perpendicular to a radius at its outer
endpoint, then the line is __________.
In
O,
1
2 and
3 = 100°. Find the following:
20.
21.
22.
23.
In
____
P,
= 300°.
Find the following:
24.
25.
26.
AD = ______
27.
DB = ______
28.
29.
PQ = 16; OX = 6
OY = 7;
30.
X
; OX = 5
RS = _____
GH = 24; OG = _____
Find the length of a chord that is 3 cm from the center of a circle with radius 9 cm.
Geometry
144
Unit 10 Worksheet 7
Solve for x. Show all work.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
(cont)
Geometry
145
Unit 10 Worksheet 7
12.
13.
14.
15.
16.
17.
18.
19.
20.
x2 + (y – 4)2 = 72
21.
Center = ________
Radius = ________
Center = (5, –1), radius =
Write the equation for the circle
given the above information.
____________________________
Geometry
146
Unit 10 Review
In problems 1 – 17 select the correct multiple choice response
1.
2.
3.
EF and EG are tangents to the circle
shown. Find the measure of
.
A.
30°
B.
15°
C.
60°
D.
12°
HJ is tangent to P.
Find the value of x.
A.
18
B.
24
C.
25
D.
32
Find
A.
B.
C.
D.
in
Find the value of x.
A.
8
B.
15
C.
22
D.
17
7.
In P the
Find
A.
42°
B.
48°
C.
96°
D.
84°
8.
WY is tangent to the circle shown.
= 74°. Find
.
A.
286°
B.
148°
C.
212°
D.
323°
9.
Find the value of x in the circle shown.
A.
54°
B.
27°
C.
49°
D.
76°
10.
Find the value of x in the circle shown.
A.
36°
B.
52°
C.
88°
D.
46°
P
290°
178°
288°
292°
4.
Find
A.
B.
C.
D.
5.
Find the value of x.
A.
40°
B.
20°
C.
60°
D.
30°
Geometry
6.
.
166°
173°
152°
208°
147
42°.
11.
What is the length of
A.
5
B.
7
C.
12
D.
14
12.
Find CD.
A.
2
B.
24
C.
9
D.
18
13.
In the circle shown, UP = 2, NP = 4,
and UW = 18. Find LP.
A.
9
B.
8
C.
12
D.
13
14.
15.
16.
Find
A.
B.
C.
D.
? 266
17.
Find
A.
B.
C.
D.
52°
78°
64°
32°
Solve the following problems. Show all work.
18.
BC and AC are tangents to the given
circle. Find the
value of x.
in
19.
If
20.
In
= 110°, find
O.
78°
90°
156°
34°
State the radius of the circle whose
equation is (x – 1)2 + (y – 3)2 = 4
A.
4
B.
16
C.
8
D.
2
State the center of the radius of the
circle whose equation is
(x + 6)2 + (y – 7)2 = 1
A.
(6, 7)
B.
(6, –7)
C.
(–6, 7)
D.
(–6, –7)
a.
b.
Q
= 220°
Find
Find
21.
Find the value of x.
22.
Circle O is inscribed in
quadrilateral ABCD. B
AB = 12 and CD = 13. 4
Find the perimeter
of quadrilateral ABCD.
6
C
D
A
Geometry
148
Unit 10 Worksheet Arc, Central Angles, and Chords
Problems 1-4 refer to
O. Find the measure of each arc.
1.
= _______
2.
= _________
3.
= _______
4.
= ________
Find the value of x. Each angle shown is a central angle.
5.
6.
x = ______
7.
x = ______
x = ______
8.
At 10 o’clock the hands of a clock form an angle of ________° .
9.
At seven o’clock the hands of a clock form an angle of ______° .
10.
If the hands of a clock form an angle of 30°, the time is ______ o’clock.
In problems 11 – 16,
11.
EB = _______
is a diameter of
12.
O.
OB = ________
13.
= _____
14.
= ______
15.
= _____
16.
DE = ________
Complete the following:
17.
AB = 8, CD = 9 ED = ______
18.
HI = _______
19.
20.
CD = _________
WY = ________
Geometry
149
Unit 10 Circles & Special Right Triangles
1.
Find AB
2.
Find AB
C
3.
is tangent to
Find AB
C
A
5
A
B
4.
Find the value of x, y, and
7.
Find JK
10.
Find
Geometry
8.
A
B
5.
OT = 9, RS = 18,
Find OR
6.
= 90° and XZ = 13
Find XY
Find LM and LO
9.
11.
12. Find SU and
Find BK and
150
Find