Download Geometry A Semester Exam Review

GEOMETRY A
Semester Exam Review
Geometry A
Semester Exam Review
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GEOMETRY A
Semester Exam Review
Right Triangles
Pythagorean Theorem
In right triangle ABC with right angle at point C:
2
2
a b  c
2
A
c
b
B
C
a
Trigonometry
In a right triangle with acute angle A:
sin A 
side opposite  A
hypotenuse
cos A 
side adjacent to  A
hypotenuse
tan A 
side opposite  A
side adjacent to  A
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GEOMETRY A
Semester Exam Review
Items on this review are grouped by Unit/Topic
Unit 1, Topic 1
1.
Write the three undefined terms in geometry.
2.
Construct the angle bisector of angle A below.
A
3.
Construct a 45o angle using point B as the vertex of the angle.
B
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Review Page 1
GEOMETRY A
4.
Semester Exam Review
A road construction crew wishes to construct a new road parallel to the current road, and
passing through the town of Herkimer, marked by point H below. Construct the location
of the road
current road
H
5.
The perpendicular bisector of AB is constructed. What is true about every point on that
perpendicular bisector?
6.
The angle bisector of CDE is constructed. What is true about every point on the angle
bisector?
7.
A line is constructed parallel to a given line. What is true about the lines?
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GEOMETRY A
Semester Exam Review
Unit 1, Topic 2
8.
Point A  1, 4  is to be transformed to point A using the translation rule
 x, y    x  3, y  6  . What are the coordinates of point
9.
A ?
Triangle ABC has been transformed to triangle ABC  .
y
A
C
A
B
O
x
C
B
a.
State in words, the transformation(s) that produce triangle ABC  .
b.
Write a function rule that represents this transformation.
c.
Why must  ABC   ABC  ?
d.
On the coordinate plane above, sketch the reflection of  ABC across the x-axis.
Label the triangle DEF. Write the function rule for this transformation.
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GEOMETRY A
10.
Semester Exam Review
Let  3,5  be a point on the coordinate plane as shown below.
y
 3,5 
O
x
Write the coordinates of the image point if  3,5  undergoes the following
transformations.
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a.
Reflected across the x-axis.
b.
Reflected across the y-axis.
c,
Translated seven units to the right and two units downward.
d.
Rotated clockwise 90 degrees about the origin.
e.
Rotated 180 degrees about the origin.
f.
Rotated counter-clockwise 90 degrees about the origin.
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GEOMETRY A
Semester Exam Review
11.
y
O
A
D
B
x
C
a.
Reflect the figure above across the line y  1 . Name the image ABC D .
b.
Does this transformation preserve lengths and angle measurements? Justify
your answer.
c.
If the figure was reflected across the x-axis, then translated two units upward,
would the result be the same as the transformation in part a) ?
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GEOMETRY A
Semester Exam Review
12.
Write the three rigid transformations.
13.
If a figure undergoes a rigid transformation, then the transformed figure must be
___________________ to the original figure.
14.
Look at the figure below.
y
O
x
Complete each statement such that each transformation will map the triangle onto itself.
a.
A reflection across the _____ axis or the line x  _____ .
b.
A rotation of _______ degrees about the point _________.
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GEOMETRY A
15.
Semester Exam Review
Triangle ABC undergoes a transformation to produce triangle ABC  . The triangles are
shown below.
y
A
C
B
O
x
B
C
A
a.
Is the transformation above a single reflection? Justify your answer.
b.
Is the transformation above a translation? Justify your answer.
c.
Determine a transformation with three reflections that will produce triangle
ABC  .
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GEOMETRY A
Semester Exam Review
Unit 1, Topic 3
16.
If figure ABCD is congruent to figure EFGH, write all eight side and angle congruence
statements.
17.
If all corresponding sides of two figures are congruent, and all corresponding angles of
the two figures are congruent, then the figures must be ________________.
In the figure below, point S is the midpoint of WU . Therefore WS  US .
R
W
T
S
U
You wish to prove  WSR   UST .
18.
To prove this congruence by SSS, what two additional congruence statements are
needed?
19.
To prove this congruence by SAS, what two additional congruence statements are
needed?
20.
To prove this congruence by ASA, what two additional congruence statements are
needed?
21.
To prove this congruence by AAS, what two additional congruence statements are
needed?
22.
State two additional congruence statements that will be insufficient to prove the
triangles congruent.
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GEOMETRY A
23.
Semester Exam Review
Consuela wants to determine the length of a power line that will be stretched over a
lake. She cannot walk through the lake. She was able to take some measurements,
hoping to determine the length. Her measurements are shown below.
Figure NOT drawn to scale
625 ft
450 ft
500 ft
500 ft
450 ft
Power Line
Consuela believes that the length of the power line is 625 feet, but she’s not sure how to
explain this to her boss.
Using what you know about triangle congruence, help Consuela by writing a brief report
as to why the length of the power line is 625 feet.
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GEOMETRY A
Semester Exam Review
Look at triangles ABC and FED below.
E
B
A
C
D
F
In items 24 through 29 below, information is given about the two triangles. State
whether the triangles can be proven congruent by ASA, SSS, AAS, or SAS. If the
triangles cannot be proven congruent, state why.
24.
AB  FE , AC  FD, A  F .
25.
A  F , B  E , C  D .
26.
A  F , B  E , AB  FE .
27.
AB  FE , AC  FD, BC  ED .
28.
B  E , AC  FD, BC  ED .
29.
A  F , AB  FE , C  D .
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GEOMETRY A
Semester Exam Review
Unit 1, Topic 4
30.
Every quadrilateral that is a parallelogram has certain properties. List all these
properties.
31.
What properties does a rectangle have in addition to those of a parallelogram?
32.
What properties does a square have in addition to those of a rectangle?
33.
Triangle DEF is isosceles with  D   E . Which sides are congruent?
34.
In the figure to the right,
D is the midpoint of AB and E is the midpoint of AC .
Figure NOT drawn to scale
Determine the values of x, y, and z.
22
C
B
10
x
y
D
E
z
12
A
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GEOMETRY A
35.
Semester Exam Review
In the figure below, D is the midpoint of AB and E is the midpoint of AC .
C
B
D
E
A
Complete the statements using the items in the box on the right. Each item may be used more
than once or not at all.
a.
b.
AD  _______
AE  _______
c.
_______ is parallel to _________
d.
AD  _______
e.
AE  _______
f.
_______ = 2  _______
g.
The ratio AB : AD is ________
h.
The ratio AE : AC is ________
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AD
AE
DE
BC
EC
BD
DE
AD
BC
EC
BD
AE
2 :1
1: 2
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GEOMETRY A
36.
Semester Exam Review
If two parallel lines are cut by a transversal:
a.
Which angle pairs are congruent?
b.
Which angle pairs are supplementary?
p
37.
Given: m  n,  1   16
1 2
3 4
Prove: p  q
9 10
11 12
38.
Given:
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7
5 6
8
m
13 14
15
n
16
D
A
 A  D
C
AB  DE
Prove: CE  CB
q
B
E
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GEOMETRY A
Semester Exam Review
D
A
C is the midpoint of BE.
39.
Given: AB  DC
AB  BE , DC  BE
Prove:  A   D
B
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C
E
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GEOMETRY A
Unit 2, Topic 1
Semester Exam Review
Figure NOT drawn to scale
30
C
B
40.
In the figure to the right, DE  BC .
a.
Prove that  ADE ~  ABC .
15
x
y
D
E
9
12
A
b.
Determine the value of x.
c.
Determine the value of y.
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GEOMETRY A
41.
Semester Exam Review
Jory wants to measure the length, L, of a pond. The figure below shows the
measurements she will use to determine the length of the pond.
In the figure below, AE and BD intersect at point C.
30 ft
A
B
25 ft
20 ft
C
40 ft
50 ft
D
Figure NOT drawn to scale
L
E
 ABC EDC ?
a.
Which similarity postulate/theorem may be used to prove
b.
What is the length, L, of the pond? Show how you determined your answer.
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GEOMETRY A
42.
43.
Semester Exam Review
If two figures are similar, then
a.
Corresponding sides are _________________________(proportional/congruent).
b.
Corresponding angles are ________________________(proportional/congruent).
In the figure below, QRS ~ QTU .
S
Q
U
R
T
a.
Prove that RS  TU .
b.
Write all congruences and proportions using the sides and angles in the figure.
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GEOMETRY A
44.
Semester Exam Review
The figure below shows  BCD .  BC D will be the image of  BCD after a dilation
with center B and scale factor 5.
Item Bank
B
C
B
C
D
Complete the statements using the item bank on the right.
D
5
1
5
parallel to
a.
The point B will be the same point as point _____.
on the same line as
b.
C D will be _______________ CD .
perpendicular
c.
BC  will be _______________ BC .
d.
The perimeter of  BC D will be _____ times the perimeter of  BCD .
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GEOMETRY A
45.
Semester Exam Review
On the graph below, quadrilateral ABCD has been dilated, with the center of dilation the
origin, to create quadrilateral ABC D .
y
10
B
9
8
7
6
B
5
A
4
3
A
2
1
-10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1O
D
D
-1
1
-2
-3
-4
2
3
4
5
6
7
8
C
9 10
x
C
-5
-6
-7
-8
-9
-10
a.
What is the scale factor of the dilation?
b.
What is the ratio of the length of AB to the length of AB ?
c.
How are the measures of  A and  A related?
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GEOMETRY A
46.
Semester Exam Review
Triangle ABC is shown on the coordinate grid below.
y
10
9
8
7
6
5
4
3
2
1
-10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 O
C
A
1
2
3
4
5
6
7
8
9 10
x
-1
-2
-3
-4
B
P
-5
-6
-7
-8
-9
-10
Triangle ABC is to be dilated by a scale factor of three with the center of dilation the
point P  5, 6  to produce triangle ABC  .
a.
On the coordinate plane above, sketch  ABC  .
b.
What is the ratio of the perimeter of  ABC to the perimeter of  ABC  ?
c.
What is the relationship between  A and  A ?
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GEOMETRY A
47.
Semester Exam Review
Pentagons ABCDE and FGHIJ are similar.
B
A
Figure NOT drawn to scale
F
10
C
J
G
15
E
D
I
12
x
H
What is the value of x?
48.
Derrick is creating a model of the Eiffel Tower. The Eiffel Tower is 301 meters tall and
has a base of width 100 meters. His model will have a base of 10 cm. What will be the
height of the model?
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GEOMETRY A
49.
Semester Exam Review
Jack wishes to find the height of a tree. He puts a stick into the ground and uses the sun’s
shadow to take some measurements. The figure below shows his measurements.
Figure NOT drawn to scale
h ft
5 ft
135 ft
9 ft
a.
Determine the height of the tree.
b.
Determine the distance from the tip of the sun’s shadow to the top of the tree.
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GEOMETRY A
Semester Exam Review
50.
a.
Draw examples of two triangles that are similar by SSS Similarity.
b.
Draw examples of two triangles that are similar by AA Similarity.
c.
Draw examples of two triangles that are similar by SAS Similarity.
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GEOMETRY A
Semester Exam Review
Unit 2, Topic 2
51.
Given right  ABC , use the item bank to identify the following. Items may be used
more than once.
A
B
C
Item Bank
A
B
C
BC
AC
BC
AB
AB
BC
AB
AC
AB
AC
AC
BC
BC
AC
AB
a.
Leg opposite  A
h.
Tangent ratio of  C
b.
Leg opposite  C
i.
The angle whose Sine ratio is
c.
Sine ratio of  A
j.
The angle whose Tangent Ratio is
d.
Cosine ratio of  A
k.
Hypotenuse
e.
Tangent ratio of  A
f.
Sine ratio of  C
g.
Cosine ratio of  C
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BC
AC
AB
BC
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GEOMETRY A
52.
Semester Exam Review
Let ABC be a right triangle, with right angle at C.
Which statements are true? Check all that apply.
A
_____  A and  B are complementary.
_____ m A  90o  m B
B
sin A
_____ tan A 
cos A
2
2
_____  AB    AC    BC 
C
2
_____ sin A  cos B

_____ cos A  sin 90o  A

_____ If  A   B, then sin A  sin B
53.
Look at the two triangles below.
16
20
12
12
20
A
16
B
Saundra says that since the two triangles are congruent that sin A  sin B . Is Saundra
correct? Justify your answer.
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GEOMETRY A
54.
Semester Exam Review
Look at the figure below.
Figure NOT drawn to scale
Q
12
R
8
S
6
10
15
T
15
U
a.
Show that the value of cos S is the same if  SQU or  SRT is used.
b.
What is the value of sinU ?
c.
Is cos U  cos T ? Why or why not?
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GEOMETRY A
55.
Semester Exam Review
A skateboarder has a piece of plywood that she wants to use as a ramp. The plywood is
16 feet long. She wants to put a vertical support under the ramp so that the ramp is at a
20o angle to the ground. This is shown in the figure below. The view is from the side of
the ramp.
16 ft. ramp
Vertical
support
Figure NOT drawn to scale
20o
How high is the vertical support? Round your answer to the nearest hundredth of a foot.
56.
Josh has a 17 ft. ladder. He would like to place the ladder against the side of a vertical
wall so that the top of the ladder is 15 feet up the wall. Safety guidelines state that
the angle that the ladder makes with the horizontal can be no more than 70 degrees.
a.
Can Josh safely place the ladder against the wall?
b.
Using the safety guidelines, what is the highest distance that the top of the 17-foot
ladder can be above the ground? Round your answer to the nearest hundredth of
a foot.
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GEOMETRY A
57.
58.
Semester Exam Review
A plane takes off from a runway and climbs at an angle of elevation of 15o.
a.
After the plane has travelled one mile (5280 feet) how far has the plane travelled
horizontally? Round your answer to the nearest foot.
b.
After the plane has travelled one mile (5280 feet) how far has the plane travelled
vertically? Round your answer to the nearest foot.
A farmer wishes to make separate sections of a piece of land as shown in the figure
below. All triangles are right triangles.
65 m
Figure NOT drawn to scale
20 m
x
y
15 m
a.
What is the value of x? _______
b.
What is the value of y? _______
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GEOMETRY A
59.
Semester Exam Review
A swimmer tries to swim straight across a stream. The stream is 100 feet wide. A current
in the stream, perpendicular to his intended path, caused him to land 20 feet downstream
from his intended landing point.
Figure NOT drawn to scale
20 ft.
Intended path
Actual path
100 ft

a.
How far did the swimmer actually swim? Give your answer to the nearest tenth
of a foot.
b.
At what angle,  , was his actual path to the intended path? Give your answer to
the nearest tenth of a degree.
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