Download Honors Geometry KEY

Honors Geometry KEY
Review Exercises for the January Exam
Here is a miscellany of exercises to help you prepare for the semester examination. You should
also use your class notes, homework, quizzes, and tests for more exercises.
Write one of the words SOMETIMES, ALWAYS, or NEVER to complete each statement. You
need to think of all possibilities to decide which word is correct.
1.
The exterior angles of a hexagon _____always____ sum to 3600.
2.
Three points are ____sometimes____collinear.
3.
Three points are ______always_____ coplanar.
4.
Parallel lines are ________ always ___ coplanar.
5.
Through two points, there is ______ always ____ exactly one line.
6.
If two rays share a common endpoint, they are ___sometimes____ opposite rays.
7.
A right triangle ____always__ has a pair of complementary interior angles.
8.
If two angles of a quadrilateral have measure 1000 and 800, then the other two interior
angles are ____never___ complementary. (Note: The other two angles, in this case, are
always supplementary.)
TRUE or FALSE?
9.
If two lines are perpendicular to the same line, then they are parallel to each other.
(FALSE: If they are non-coplanar, they are not necessarily parallel. If all lines are
coplanar, this is a true statement.)
10.
In an orthographic drawing of a solid made of cubes, the right view and the left view are
identical. (FALSE. Ignore this question!)
11.
If parallel lines are cut by a transversal, then alternate interior angles could be
complementary. (TRUE: The importance of the word “could” is that it is not as
“strong” as IS. This does not make the statement false.)
12.
The supplements of congruent angles are congruent. (TRUE)
13.
Perpendicular is a symmetric relation. (TRUE: This says that if a  b, then b  a)
14.
“Greater than” is transitive. (TRUE)
15.
Supplementary angles must be adjacent. (FALSE)
Honors Geometry Exam Review Questions
16.
page 2
Each interior angle of a regular hexagon has a measure of 1200. (TRUE, using
(n  2)180
with n = 6.)
n
Complete the following.
17.
An __angle__ is the union of two rays with a common endpoint.
18.
Perpendicular lines form _____right angles__________.
19.
If parallel lines are cut by a transversal, then ___same side interior angles______ are
supplementary. (Another answer is “same side exterior angles”)
20.
In a conditional statement, if you reverse the hypothesis and conclusion, you get the
_______converse______ of the original statement.
21.
If an original conditional statement is false, then the ____contrapositive____ of the
original statement must also be false.
22.
Examine the diagram.
F
P
A
Q
B
D
C
E
23.
Name the following:
f) The intersection of planes P and Q BD
g) A pair of skew lines BD and FC

h) The intersection of plane Q and FC C
Refer to the diagram below.
a) If mPXR = 42,find the measure of these angles:
mRXT
mQXS
mSXP
480
420
1380
P
R
T
X
S
True or False
a) A, B, D, and E are coplanar true
b) B and C are collinear true
c) A, F, and C are collinear true
d) F lies in plane P false

e) FC intersects plane P at point A and plane Q at
point C true
Q
b) If mSPX = 22 and PX = SX, find mPXS.
= 1800 – 440 = 1360
c) If RPX  SQX , then what lines must be parallel? Give
the reason. SQ || PR b/c AIC  P
d) If SXQ  TQX, then what lines must be parallel?
Give the reason. SX || TQ b/c AIC  P
Honors Geometry Exam Review Questions
24.
25.
page 3
Write three postulates. Some samples are provided. There are others.
a.
Through two points, there is one line.
b.
If two parallel lines are cut by a transversal, then corresponding angles are
congruent.
c.
If two planes intersect, then their intersection is a line.
Write three theorems.
a.
In a triangle, the sum of the interior angles is 180 degrees.
b.
If two lines are cut by a transversal, then the alternate interior angles are
congruent.
c.
Vertical angles are congruent.
Solve for any variable in the drawings.
26.
27.
28.
29.
30.
31.
Honors Geometry Exam Review Questions
32.
28-gon
page 4
33.
each interior angle = ________
34.
M is the midpoint of CD . The coordinates are:
C (6 , x)
M (1 , x – 4)
D (-4 , 11)
35.
Find the length of CD if C (6 , -3) and ( 0 , 8).
36.
Where is the midpoint of AQ if A(9 , 4) and Q(-1, 11) ?
37.
A line passes through the points A( -4, 5) and B(2, -7).
(5  7) 2  (4  2) 2  144  36  180  6 5
a)
Find AB =
b)

5  7 12
Find the slope of AB =

 2
4  2 6
c)

Find an equation of AB . Write your answer in point-slope form.
y + 7 = -2(x – 2) OR y – 5 = -2(x + 4)
d)
 4  2 5  7 
Find the coordinate of the midpoint of AB M 
,
= (-1,-1)
2 
 2
Honors Geometry Exam Review Questions
page 5
Construct these with compass and straightedge. These are done approximately, since I do not
have compass and straightedge that works on the tablet.
38.
an isosceles right triangle with sides (non-hypotenuse) given.
39.
a line perpendicular to AB through C.
40.
a square with side EF.
41.
a 450 angle
42.
the complement of < A
Honors Geometry Exam Review Questions
page 6
Draw each of these.
43.
This isometric drawing has a volume of 9 cubes. Make a drawing with cubes whose
volume is 13 cubes.
\
44.
Calculate x.
45.
x + 73 + 73 = 180
x = 34
46.
Calculate x.
x + x + 26 = 180
2x = 154
x = 77
Write “Pizza is served if and only if today is Tuesday” as two “if …then …”
statements.
If pizza is served, then today is Tuesday.
served.
and
If today is Tuesday, then pizza is
Write the “only if” half as an “if …then …” statement.
If pizza is served, then today is Tuesday
Honors Geometry Exam Review Questions
47.
page 7
Consider a set on n noncollinear points. Connect each point with all other points. Count
the number of line segments.
# of segments = 3
# of segments = 6
# of segments = 10
If there are n noncollinear points, what is an expression for the total number of segments
which can be drawn?
f(n) = n +
n(n  3)
2
This is the number of sides of the polygon (n) plus the number of diagonals drawn
from each vertex times the number of vertices. The number of diagonals is divided
by two, since they were all counted twice.
48.
How many sides and diagonals (total) does a decagon have? a 27-gon? an n-gon?
From above, a decagon has 10 +
An n-gon has n +
1
1
(10)(7) = 45. A 27-gon has 27+ (27)(24) = 351
2
2
n(n  3)
sides and diagonals total.
2
How is this related to the handshake problem? It is the same question as the
handshake problem.
49.
Given: A rolling stone gathers no moss.
a)
Rewrite the statement as a conditional.
If a stone is rolling, then it gathers no moss.
b)
Write the converse.
If a stone gathers no moss, then it is rolling.
c)
Write a biconditional combining the statement and its converse.
Honors Geometry Exam Review Questions
page 8
A stone is rolling if and only if it gathers no moss.
d)

This big gray stone is not rolling, therefore it is gathering moss.
(This is using the inverse, whose truth is not known from the truth of the
original.)

That red stone is gathering moss, therefore it is not rolling. TRUE
(This is using the contrapositive, which is TRUE if the original is true.)

That green stone is not gathering moss, therefore it is rolling.
(This is using the converse, so we do not know whether it is true or false.)
Make a truth table for this logic statement: ( P  Q)  (Q ~ P)
50.
P
T
T
F
Q
T
F
T
F
F
51.
Assume the given statement is true. Which of these statements must be true?
Make a conclusion (if possible) and name the pattern of reasoning.
P  R
~ R W
( S  R)  W
a) R
 P
modus tollens
(law of
contrapositive)
52.
b) R
__ W __
law of disjunction
c) ( S  R )
__ W __
modus ponens
(law of
detachment)
PR
d) R  Q
__ P  Q __
law of syllogism
Assuming A and B are both true, P and Q are both false, and X and Y are of unknown
truth value. Determine (if possible) the truth value of each statement.
a) A  ( B  X )
Since B is true, then B  X is true (regardless of the truth value of X). So
with A being true, A  ( B  X ) is true AND true, so the entire statement is
TRUE.
b) B  ( P  A)
Honors Geometry Exam Review Questions
page 9
Starting with P  A , this conjoins a FALSE and TRUE, which is TRUE.
Then the if…then gives TRUE  TRUE, so the entire statement is TRUE.
c) ( A  X )  ( B  Q)
A  X is TRUE because A is TRUE. B  Q is FALSE because Q is FALSE.
The conditional is FALSE because TRUE FALSE leads to a false
statement.
53.
Write a logical expression which is logically equivalent to ( P  Q)  Q .
P Q
PQ
Q ( P  Q )  Q
________________________________________
T T
T
F
F
T F
T
T
T
F T
T
F
F
F F
F
T
T
This is logically equivalent to  Q
54.
Write the inverse of “People wear flip flops only if they hate real shoes.”
First, re-write this statement in an “if … then …” form.
If people wear flip flops, then they hate real shoes.
The inverse is: If people do not wear flip flops, then they do not hate real shoes.
55.
Give an example of a syllogism.
If A  B and B  C, then A  C
56.
Give an example of the law of disjunction.
If (A or B) and not B , then we
can conclude A.
57.
d
6
Which lines (if any) must be parallel?
1
a
9
5 10
a) 1  2 a||b
b) 6  7 c|| d
c) 9  4 a|| b
d) 10  11 a || b
e) 11  8 none
f) 5 and 11 are supp a|| b
g) 4 and 12 are supp c||d
h) 5 and 10 are supp none
b
2
12
c
11
3
8
4
7
Honors Geometry Exam Review Questions
58.
page 10
Assume that a || b and c ||d
d
6
m< 1 = 960
1
a
m < 6 = 400
9
5 10
Calculate all of the other numbered
angles.
b
2
12
4
Draw each of these triangles (if possible).
a)
right triangle
c)
b)
60.
8
7
c
59.
11
3
isosceles acute triangle
scalene obtuse triangle

Draw two horizontal planes, A and B, a vertical plane, C, and a line DE so that D lies in C,
and E lies in A
Honors Geometry Exam Review Questions
61.
page 11
What is the locus of the center of a sphere that rolls around on a rectangular table top
whose surface measures 5 feet by 3 feet?
The locus is a part of a plane which is 5 feet by 3 feet (a rectangle) which is above
the table top by the size of the radius of the sphere.
62.
What are the loci of points that are equidistant from two points, A and B and also
equidistant from two parallel planes?
The loci is the intersection of a plane which is equidistant from A and B and the
perpendicular bisector of AB , and a plane which is between the parallel planes. This
intersection is either a line, a plane, or the null set (no intersection).
63.
a)
What is the locus of points, in a plane, which are equidistant from two perpendicular
rays which have a common endpoint?
The locus is a line which is the bisector of the angle created by the two rays, so the
Bisector forms a 450 angle with the rays.
b)
What is the locus of points which are equidistant from two perpendicular
rays which have a common endpoint?
The locus is a plane which is the bisector of the angle created by the two rays, so the
Bisector forms a 450 angle with the rays.
Calculate the angles.
64.
2x = (x + 20) + 34
2x = x + 54
x = 54
Angles measure 108 and 74.
65.
The angles of a triangle are in the ratio 2:3:5. Find the measure of the largest
angle in the triangle. Always support your answers with appropriate algebra.
Use 2x, 3x, and 5x 2x + 3x + 5x = 180 so 10x = 180, thus x = 18
The angles measure 36, 54, and 90. So 90 is the largest angle in the triangle.
66.
Y = 73 + 81 = 154
x +y = 180, so x = 26
x – 10 = 16
z + (x – 10) = x
So z + 16 = 26, making z = 10
w + z = 180, so w = 170
67.
x + 46 + 46 = 180
x = 88
Geometry Exam Review Exercises
68.
page 13
Find the measure of each interior angle of a regular octagon.
(n  2)180 (8  2)180

 1350
8
n
69.
How many sides does a regular polygon have if each exterior angle measures 200?
360
 20
n
70.

20n = 360 
n = 18 sides
An interior angle and an exterior angle of a regular polygon differ by 1000. How
large is the interior angle?
(n  2)180 360

 100  180n-360 – 360 = 100n
n
n
180n – 100n = 720  80 n = 720  n = 9 sides
0
So the interior angles are 1400 (Supp of the 40 exterior angles).
71.
Find the sum of the measures of the interior angles of a dodecagon.
(n – 2)180 
72.
(12 – 2) 180  18000
How many sides does a regular polygon have if the interior angles are 200 more
than 3 times the exterior angles?
180(n  2)
360
3
 20  180(n-2) = 3*360 + 20n
n
n
 180n – 360 = 3*360 + 20n

160n = 1440

n = 9
73.
What is the sum of the exterior angles in a 64-gon?
3600 (Enough said!)
Geometry Exam Review Exercises
page 14
Proofs.
74.
Given: BA  BC , < 1  < 3
Prove: <2 is comp to < 3
Proof:
75.
1. BA  BC
1.
given
2.
2.
Perpendicular pairs
are comp.
given
< 1 is comp to < 2
3.
<1  <3
3.
4.
< 2 is comp to < 3
4. If an angle is comp to
one of two congruent
angles, then it is comp to
the other angle also.
Given: a || b , < 1  < 3
Prove: c || d
Proof:
1.
a ||b
1.
given
2. < 2  < 3
2.
PAEC
3. < 1  < 3
3.
given
4.
<1  <2
4.
Transitive for 
5.
c || d
5.
AIC P
Geometry Exam Review Exercises
76.
page 15
Given: < 1  < 4
Prove: < 2  < 3
Proof:
77.
1.
< 1 is supp to < 2
1. Linear pairs are supp
2.
< 3 is supp to < 4
2.
3.
<1  <4
3. Given
4.
<2  <3
4. Congruent supps thm
(If two <’s are supp
to the same < or  <’s,
then the <’s are  )
Linear pairs are supp
Given: < DAB  < DBA
<2  <3
Prove: < 1  < 4
Proof: 1.
< DAB  < DBA
1.
Given
2.
m< DAB = m< DBA
2.
Defn of  <’s
3.
m < 1 + m < 2 = m < DAB
3.
Angle addition post.
4.
m < 3 + m < 4 = m < DBA
4.
Angle addition post
5. m < 1 + m < 2 = m < 3 + m < 4
5.
Substitution
6.
<2  <3
6.
given
7.
m<2=m<3
7.
Defn of  <’s
8.
m<1=m<4
8.
Subtraction prop of =
Geometry Exam Review Exercises
9.
page 16
<1  <4
9.
Defn of  <’s
Calculate the areas and volumes.
78.
79.
Area =
1
1
bh  (8)(5) = 20
2
2
area = bh. By the Pythagorean
Theorem, the base is 12. So
area = (12)(5) = 60
80.
81.
Area = bh = (12)6) = 72
area =(8)(6) – (14+8+1.5 + 6) =18.5
Geometry Exam Review Exercises
82.
83.
Area = (6)(6) +
84.
page 17
1
(6)(9) = 36 + 27 = 63
2
1
h(b1  b2 )
2
Show why this formulas should be true.
(i.e., drive the formula)
The area of a trapezoid is A =
Area =
=
1
1
xh + b1H + h(b2 – x – b1)
2
2
1
1
1
1
1
xh + (2b1h) + hb2 - xh - b1h
2
2
2
2
2
=
1
h(x + 2b1h + hb2 – xh – b1h)
2
=
1
(b1 + b2)
2
A figure is drawn on lattice paper
(dot paper) with 4 interior points and
11 points on its boundary. What is
its area. Pick’s Theorem says that
1
the area equals (boundary
2
points) + interior points – 1, so the
area is .5(11)+ 4 - 1 = 8.5