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Basic Geometry
Exam Date _________ Time _____
Final Exam Review
Chapter 5
5-1 Bisectors, Medians, & Altitudes
Fill in the blank and match the name of each special segment to the correct diagram.
1) perpendicular bisector – segment that is
____________________ to the opposite
side of a triangle and also ____________
the opposite side.
a)
2) altitude – segment that is ________________
to the opposite side of a triangle.
b)
3) angle bisector – segment that _____________
an angle of a triangle.
c)
4) median – segment that ______________ the
opposite side of a triangle.
d)
Use the given diagram to solve for the variable and missing measures.
5) If MS is the altitude of D MNQ, and m
and m 2 = 7x + 9, find the following:
1 = 3x + 11,
x = _____
m 2 = ______
1
6) If MS is the median of D MNQ, and QS = 3x – 14,
SN = 2x + 1, and m MSQ = 7x + 1, find the following:
x = _____
SN = _____
QN = _____
Is
MS also an altitude of D MSQ? Explain.
7) If WP is an angle bisector of D WAH and m HWP = x + 12,
m PAW = 3x – 2, and m HWA = 4x – 16, find the following:
x = ______
m
HWP = ______
m
PAW = ______
8) If WP is the perpendicular bisector of D WAH,
and m WHA = 8x + 17, m HWP = 10 + x,
AP = 6y + 4, and PH = 22 + 3y, find the following:
x = ______
m
HWP = ______
y = ______
AH = ______
2
5-2 Inequalities & Triangles
9) Name the longest side of each triangle.
a) ____________
b)_____________
A
c)______________
X
78°
M
104°
C
42°
Z
Y
N
P
B
10) Name the largest angle in each triangle.
a) ____________
b) ____________
A
J
15
c) _____________
12
10
P
K
5
B
18
C
Q
13
12
14
L
R
11) Find the value of x. List the sides of D PQR in order from shortest to largest for
the given angle measures. (Hint: Draw a diagram)
m P = 9x + 29, m Q = 93 – 5x, m R = 10x + 2
________, _______, _______
3
5-4 The Triangle Inequality
12) Can a triangle be formed from the given lengths?
a) 12 in, 16 in, 20 in
____________
b) 2.4 ft, 5.7 ft, 8 ft
____________
c) 28¼ m, 43⅜ m, 71⅝ m
___________
13) The measures of 2 sides of a triangle are given. The measure of the third side
must be between what 2 measures?
a) 16 ft, 17 ft
____________
b) 8¾ in, 8¾ in
____________
c) 9.2 cm, 17.8 cm
____________
14) Determine whether the given coordinates are the vertices of a triangle. Explain.
A(5, 8), B(2, -4), C(-3, -1)
AB = _____
BC = _____
AC = _____
Using the Pythagorean Theorem
15) Tell if the triangle with the given lengths is acute, obtuse, or right.
a) 6, 7, 10
____________
b) 11, 11, 5
____________
c) 1, 2,
5
___________
4
Chapter 6
6-1 Ratio & Proportions
1) Solve each proportion.
a)
13 26
=
49 7 x
b)
2y - 12 -4
=
28
7
c)
k +1 5
=
k -1 6
2) Solve each problem using a proportion or writing an equation.
a) A 36 inch ribbon is cut into pieces so
that the shorter piece and the longer piece
have a 2:6 ratio. Find the length of the
longer piece of ribbon.
b) The lengths of a triangle are in a ratio of
2:3:5. If the perimeter of the triangle is
120 centimeters. Find the measure of each
side of the triangle.
c) The measures of the angles of a triangle
have a ratio of 2:3:4. Find the measure of
the largest angle.
c) The ratio of the length of a rectangle to
its width is 4:7. If the perimeter is 176,
find the dimensions of the rectangle.
5
6-2 Similar Polygons
3) State the 2 conditions for 2 polygons to be similar.
_______________________________________________________
_______________________________________________________
4) Determine if the given figures are similar. State why or why not.
If so, state the scale factor and state the similarity statement.
a)
b)
5) Rectangle ABCD is similar to rectangle ATSU.
a) What is the scale factor of ABCD to ATSU? _______
T
b) Find the ratio of the perimeter of
rectangle ATSU to that of ABCD. 5.4
S
C
B
3
c) Find the ratio of the area of
rectangle ATSU to that of ABCD.
U
A
D
8
d) How is the scale factor related to the ratio of the perimeter of a polygon?
e) How is the scale factor related to the ratio of the area of polygons.?
6
6) The two given polygons are similar. Find x and y.
a) x = _____
y = _____
b) x = _____
c) x = _____
y = ___ __
d) x = _____
y = _____
y = _____
7) Fill in the blank with sometimes, always, or never.
a) Two squares are ______________________ similar.
b) Two rectangles are _____________________ similar.
c) A scalene triangle and an isosceles triangle are __________________ similar.
d) Two regular polygons are ____________________ similar.
7
6-3 Similar Triangles
8) State the 3 triangle similarity theorems. __________, _________, __________
9) State if each pair of triangles is similar. State the triangle similarity theorem and
the triangle similarity statement.
a) ________
b) ________
c) ________
6-4 Proportions & Similarity in Triangles
10) Find the indicated variables.
a) _________
b) __________
c) __________
9
7
3
y
k
8
Chapter 7
7-2 The Pythagorean Theorem & its Converse
1) Can the given numbers be the lengths of the sides of a right triangle? Why or why not?
a) 7, 13, 18
b) 10, 24, 26
c)
12 ,
13 , 5
2) Which of the following figures are rectangles? Why or why not?
a) _______
17
b) _______
8
11
4
c) _______
4
20
4 2
16
12
7-3 Special Right Triangles
3) Show how to represent the sides for the following special triangles.
45°, 45°, 90° triangle
x
30°, 60°, 90° triangle
60
x
30
9
4) Find the value of the variables.
b) x = _____ y = _____
a) x = _____ y = ______
45
y
45
y
60
x
17
x
30
5 3
d) x = _____ y = _____
c) x = _____ y = _____
30
x
60
26
y
x
30
60
y
4 5
f) x = _____
e) x = _____ y = _____
y
x
y = _____
60
x
11
30
y
9
g) x = _____ y = _____
60
h) x = _____ y = _____
30
9
y
x
y
x
45
45
8
10
5) Draw a diagram for each problem, then answer the question.
a) The perimeter of a square is 28 cm.
What is the length of the diagonal?
b) The diagonal of a rectangle divides the
rectangle into two 30, 60, 90 triangles. Find
the dimensions of the rectangle, if the
length of the diagonal measures 18 inches?
7-4 Trigonometry
6) Find the following trig ratios and express each value as a decimal rounded to the
nearest thousandth:
B
sin A = _____ = ______
sin B = _____ = ______
cos A = _____ = ______
cos B = _____ = ______
tan A = _____ = ______
tan B = _____ = ______
5
A
1
2
C
7) Find the value of each angle. Round to the nearest degree.
a) sin A = 7
11
m<A = _____
b) cos A = 12
23
m<A = _____
c) tan A = 1.483
m<A = _____
d) sin A = 0.995
m<A = _____
e) cos A = 0.743
m<A = _____
f) tanA = 0.194
m<A = _____
11
8) Find the missing length or angle for each problem. (Round to tenths)
a) x = _____
b) x = _____
21
x
68°
9
x
32°
d) x = _____
x°
c) x = _____
x
54°
6
e) x = _____
14
x
f) x = _____
75°
18
g) x = _____
h) x = _____
x°
4.6
6.8
i) x = _____
29°
12.4
x
8
24
10.6
33.4
x°
83°
x
12
7-5 Angles of Elevation & Depression
9) Write an equation to represent each problem and solve.
a) The ground crew for a hot-air balloon can see the
balloon in the sky at an angle of elevation of 11°. The
pilot radios to the crew that the hot-air balloon in
950 feet above the ground. Estimate the horizontal
distance of the hot-air balloon from the ground crew.
b) Kierra is flying a kite. She has let out 55 feet of
string. If the string makes a 35° angle with the
ground, how high above the ground is the kite?
Draw diagram here.
c) A 12 foot ladder leans against a building and makes
a 78º angle with the ground. How far from the
building is the base of the ladder?
Draw diagram here.
d) At a certain time of the day, the angle of
elevation of the sun is 44º. Find the length of the
shadow cast by a building 30 meters high.
Draw diagram here.
13
Chapter 8
8-1 Angles of Polygons
1) Find the sum of the interior angles of each convex polygon.
a) heptagon _______
b) 37-gon _________
c) nonagon _______
2) What is the measure of each interior angle for the given regular polygons?
a) octagon _______
b) decagon ________
c) 90-gon _______
3) The measure of each exterior angle of a regular polygon is given. Find the number of
sides of the polygon.
a) 72º
______
b) 45º
_____
c) 14.4º _____
4) What is the sum of the exterior angles of a decagon?_________
5) What is the measure of each exterior angle in a regular 18-gon? _________
6) How many sides does a regular polygon have if each interior angle measures 172.5º?______
7) Find the value of x.
a) x = ________
b) x = _______
c) x = ________
x°
88°
110°
87°
x°
77°
x°
93°
x°
73°
102°
50°
14
d) x = ________
e) x = ________
x°
108°
2x°
x°
123°
f) x = _________
x°
55°
120°
135°
g) x = ________
h) x = ________
i) x = _________
(regular octagon)
x°
x°
44º
115°
76°
x°
x°
8) True or False.
a) Every quadrilateral is a parallelogram. ______
b) If both pairs of opposite angles are congruent, then the quadrilateral is a
parallelogram. ______
c) The diagonals of a trapezoid are congruent. ______
d) Every square is a rhombus. ______
e) All rectangles are squares. ______
15
Properties of Parallelograms and Other Quadrilaterals
9) Find each length or angle measure.
a) ABCD is a rectangle
A
E
b) PQRS is a parallelogram
B
P
Q
28°
39°
T
20
76°
D
48
C
S
R
AD = ______
m
PSR = _______
AC = ______
m
SRQ = _______
BE = ______
m
PST = _______
m
m
PTQ= _______
ABC = ______
c) DEFG is a rhombus
E
D
d) KITE is a kite
I
12
S
H
8
F
K
6
G
16
13
FG = ______
E
ET = _______
FD = ______
KI = _______
EH = ______
KS = _______
m
DHG = _______
T
m
IST = _______
16
10) Find the value of the variables for each parallelogram.
a) x = _____ y = _____
b) x = _____ y = _____
c) u = _____ v = _____
d) b = _____ c = _____ d = ______
11) Find the value of the given variable.
a) x = ______
b) x = ______
x
17
Chapter 11
Area of Triangles, Quadrilaterals, Regular Polygons, & Circles
1) Find the area of each polygon.
a) A = _________
b) A = __________
5
17
c) A = __________
6
8
25
11
d) A = ________
23
25
e) A = __________
f) A = __________
15.2
10
4.5
6
10
2) Find the area of each irregular figure.
a) A = _______
b) A = ________
6
5
6
8
5
21
5
8
6
8
18
14
18
3) Find the area of the shaded region.
a) A = ________
b) A = ________
5
8
5
7
5
5
16
Chapter 12
Surface Area & Volume
1) Find the base area, lateral area, surface area, and volume of each figure.
a) B = _______
b) B = ________
c) B = ________
LA = _______
LA = ________
LA = ________
SA = _______
SA = ________
SA = ________
V = _______
V = ________
V = _______
19
d) B = _______
e) B = ________
f) B = ________
LA = _______
LA = ________
LA = ________
SA = _______
SA = ________
SA = ________
V = _______
V = ________
V = _______
20