Level 1 Geometry Review 7: Angles and Polygons
Polygon angle formulas - Let n = the number of sides in a polygon.
The sum of the interior angles in any n-gon:
(n – 2) 180
( n  2)180
One interior angle in a regular n-gon:
n
The sum of the exterior angles in any n-gon:
360
360
The measure of an exterior angle in a regular n-gon:
n
_______________________________________________________________________
1. Define a regular polygon.
2. What is the sum of the interior angles of a 40 sided polygon?
3. Given the angles shown in this polygon, find the value of x.
119
105
2x
3x
2x
90
4. Given a regular 90-gon how large is each exterior angle?
5. What is the size of each interior angle of a regular dodecagon (12 sided polygon)?
6. Suppose a regular polygon has exterior angles of size 0.5º. How many sides are there?
7. Suppose a regular polygon has each interior angle of size 170º. How many sides are
there?
8. Suppose a polygon’s interior angles add to 7200º. How many sides are there?
9. True or false? A polygon’s interior angles can add up to 1970º.
10. In a regular polygon each interior angle is 170º larger than each exterior angle.
How many sides are there?
11. The supplement of an angle is 6.5 times as large as the angle. Find the angle.
12. The angles of a triangle are in the ratio of 5:5:6. Find the angles.
13. The complement of an angle is 22º more than the angle. Find the angle.
14. Twice the supplement of an angle is 230º. Find the angle.
15. Half the complement of an angle when added to one quarter of the angle makes 32.5º.
Find the angle and its complement.
16. Two sides of a triangle are of lengths 5 and 14. What are all the possible lengths for
the third side?
Level 1 Geometry Review 8: Congruent Triangles
1.
2.
3.
4.
5.
6.
7.
8.
9.
Vertical angles are congruent.
Base Angles Theorem – If 2 sides of a triangle are congruent, then the opposite
angles are congruent. And its converse – If 2 angles of a triangle are congruent
then the opposite sides are congruent.
Proving triangles congruent: ASA, SAS, SSS, AAS, and HL
Properties of parallelograms
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Diagonal bisect each other
Properties of rectangles
All properties of a parallelogram
All angles are right.
Diagonals are congruent.
Properties of a rhombus
All properties of a parallelogram
All sides are congruent.
Diagonals are perpendicular.
Diagonals bisect opposite angles.
Properties of a square
All properties of a rectangle
All properties of a rhombus
Properties of an isosceles trapezoid
One pair of opposite sides parallel
Legs are congruent.
Base angles are congruent.
Diagonals are congruent.
Besides the properties, you can prove that a quadrilateral is a parallelogram by
proving that one pair of opposite sides are both congruent and parallel.
Problems.
1. Given: ACDF is a parallelogram.
AFB = ECD
E
F
D
Prove: FBCE is a parallelogram.
A
AC  AR
C
A
2. Given: AC  BK
C  K
Prove: BARK is a parallelogram.
B
C
B
R
K
3. Given: NRTW is a parallelogram.
NX  TS
W
WV  PR
V
T
X
Prove: XPSV is a parallelogram.
S
N
4. Given: RSOT is a parallelogram.
MS  TP
P
R
P
R
T
Prove: MOPR is a parallelogram.
S
M
O
5.
Given: AB  AC
Prove: 1  2
A
2
1
B
6.
Given:
KRM  PRO
C
R
KR  PR
Prove: RM  RO
K
7.
M
SX  TY
P
O
R
Given: WX  YZ
SW  TZ
Prove: RW  RZ
T
S
W
X
Y
Z
8.
PR  ST
N
V
Given: NP  VT
P  T
Prove:
W
∆WRS is isosceles.
P
9.
Given:
FG  JH
FGH  JHG
S
R
F
T
J
K
Prove: ∆FKJ is isosceles.
G
H
Level 1 Geometry Review 9
1. A coordinate proof.
X (0, b)
Given: any isosceles triangle, positioned cleverly
as shown.
Show that the medians of XYZ from Y and Z are
congruent.
Reminders:
 A median is a line segment from a vertex
of a triangle to the midpoint of the other
side.


Midpoint formula:
x  x y  y 
1
2
2


, 1


2
2



Distance formula:
(x1  x2 )2  (y1  y2 )2

Z
(-a, 0)
Y
(0, 0)
(a, 0)
2. Given:
BED  BDE
C
A
BEA  BDC
B
AE  CD
Prove:
A  C
D
E
3. Given:
AB  AC
A
AD  AE
E
D
Prove ∆FBC is isosceles
F
B
C
Level 1 Geometry Review 4
The following are valid reasons for proving that two triangles are congruent:
ASA, AAS, SAS, SSS, HL. Other reasons are not enough: AAA, SSA
1. In each diagram state why the triangles are congruent or write “NN” (not necessarily
congruent).
Pay no attention to the appearance of the triangles, only the facts given.
a.
b.
d.
g.
2.
Given: AB = CD.
AX  BD.
CY  BD.
BX = DY.
c.
e.
f.
h.
i.
A
B
X
Prove: ABCD is a parallelogram.
Y
D
C
3. Find the value of x.
a.
20º
xº
48º
b.
19º
xº
4.
Given:
AO  OB
D
A
O
AC || BD
Prove: ∆AOC  ∆BOD
B
C
5. Find x.
a.
b.
xº
64º
48º
xº
c.
d.
2xº
5xº
xº
78º
4xº
e.
f.
80º
xº
58º
39º
xº
Level 1 Geometry Review 5
a
o
t
h
a
opposite
sin( x) 
hypotenuse
s
Hypotenuse
Opposite
X
Adjacent
o
h
c
cos(x) 
adjacent
hypotenuse
tan( x) 
opposite
adjacent
Examples:
140
x
x cos(18º )  140
140
x
cos(18º )
x  147.2
130
tan( x) 
275
130
x  tan 1 (
)
275
x  25.3º
cos(18º ) 
x
18
140
130
xº
275
Problems:
1. How high is the tree?
2. Find the height of
this parallelogram
and then its area.
22
81º
15
height =
area =
40
244 ft.
3. Find all missing
sides and angles.
Note that the third
side can be found by
the Pythagorean
Theorem or trig.
x=
y=
z=
y
x
12
z
77
4. While watching a
spaceship take off you
hear on the radio that it
is 14 miles in the air.
You have to look up at
an angle of 86º to see it.
How far from the
blastoff position are
you?
rocket
14 mi.
86
x
5. Given P, find the
area of the shaded
segment.
6. Find the perimeter of
the triangle.
16
25º
70
Area of sector =
Area of triangle =
Area of segment =
7. Find the area of this
regular pentagon by
dividing it into
triangles from the
center.
P
20
8. Without a calculator, tan(45º) =
you should be able to
find these values by cos(60º) =
drawing and labeling
familiar triangles.
sin(45º) =
83
25º
Level 1 Geometry Review 6
Area, Surface Area, Volume
1. Find the shaded area.
17
10
10
2. Find the shaded triangular area.
8
5
6
3. This figure is a parallelogram. Find length x.
12
135
10
Area = 195 sq. units
x
4. In each figure congruent
circles just fit in congruent
squares with sides of length 20.
For each one find the
percentage of the square’s area
occupied by the circles.
20
Explain your result.
5. Find the sector area.
12
80º
6. This is a semicircle. Its perimeter is 65. What is its radius?
20
20
7. A cylinder has a base area of 25π square inches
and a volume of 375π cubic inches.
Base radius =
Altitude =
Lateral area =
8. A cone has a volume of 1500 cubic inches. The
altitude is equal to the diameter of the base.
Altitude =
9. A right prism has a right triangular base. The
legs of the base are 21 and 28 and the altitude of
the prism is 25.
Volume =
Total surface area =
21
25
28
10.
x=
116º
y=
2x+10
29y
11.
x=
3x+4y
50
y=
5x
12.
x=
x
y=
2y
4x+y
Level 1 Geometry Review 1
Conditional or If-Then statements.
Suppose a certain statement “If P then Q” is true.
Original statement: If p then q. (pq)
Suppose this is true.
Converse:
If q then p. (qp)
This is not necessarily true.
Inverse:
If not p then not q. (~p~q)
This is not necessarily true.
Contrapositive:
If not q then not p. (~q~p)
Always true if the original
statement is true.
_______________________________________________________________________
1.
Assume this statement is always true:
If you bang your head against the wall you will get a headache.
a.
Make a Venn diagram for this situation.
b.
Write the converse and state whether it is necessarily true.
c.
Write the inverse and state whether it is necessarily true.
d.
Write the contrapositive and state whether it is necessarily true.
e.
Joe Max banged his head against the wall. Put Joe Max in the diagram (possibly
in more than one place) and draw any conclusion you can about him.
f.
Ozzie has never had a headache. Put Ozzie in the picture, wherever possible, and
draw any conclusion you can.
g.
Latrice has a headache. Place her and draw a conclusion if you can.
h
Charmaine has never banged her head against the wall. Place her and conclude
what you can.
2.
Assume the following to be always true statements:
A watched pot of water never boils.
To be able to cook spaghetti you must boil a pot of water.
Determine whether the following are always true, always false or neither.
a.
If you are cooking spaghetti you did not watch the pot.
b.
If you don't watch a pot of water, it will boil.
c.
If you boil water you can cook some spaghetti.
d.
If you are boiling water you did not watch the pot.
Circle theorems:
C
A
2X
D
X
B
x
r
76º
Area of a sector of
angle x in a circle of
radius r:
x
A=
š r2
360
95ª
85º
104º
3.
Find the radius of the
circle.
4.
Find the radius.
16
7
A
C
2
5
5
D
B
5. Two concentric
circles have radii of 17
and 7. The chord is
tangent to the small
circle. What is its
length?
6. C is the center of
this circle and the
triangle is inscribed in
it. Find the size of
angle X.
7. Find the missing
angles and arcs.
50º
X
C
8. Find angle X.
X
42º
28º
172º
Level 1 Geometry Review 2
128º
1.
An architect is designing a house and builds a scale model. The model is 2 feet
high and the house will be 24 feet high. How do the heights of the front doors
compare, the floor areas of the master bedrooms compare, and the volume of the
attics compare in the model to the actual house?
2.
Three regular polygons surround a point. How many sides has the largest
polygon?
3.
Find x.
150º
xº
4.
A sphere just fits in a cube whose sides are of length 10.
What percentage of the cube’s volume is occupied by the sphere?
10
Level 1 Geometry Review 3
1. Coordinate transformations.
Be prepared to recognize TRANSLATIONS, ROTATIONS, REFLECTIONS and
combinations like GLIDE REFLECTIONS.
For each problem:
i. Transform each figure according to the transformations given by applying them to the
vertices of the figure and then connecting the new vertices. Draw the new figure on the
same graph.
Here’s the notation: x1 is the starting x –coordinate of a point.
x2 is the x-coordinate of the point after the transformation.
y1 is the starting y –coordinate of a point.
y2 is the y-coordinate of the point after the transformation.
ii. Describe the transformation in words using the terminology above, if it applies.
FOR TRANSLATIONS SPECIFY HOW FAR AND IN WHAT DIRECTION.
FOR ROTATIONS SPECIFY THE CENTER AND ANGLE TURNED.
FOR REFLECTIONS SPECIFY THE LINE OF REFLECTION.
6
a.
x2  x1  5
y2  y1  3
4
Describe the transformation.
2
-10
-5
5
10
-2
-4
-6
6
b.
x2  x1
y2  2  y1
4
Describe the transformation.
2
-10
-5
5
-2
-4
-6
10
6
c.
1
x1
2
y2  4y1
x2 
4
2
Describe the transformation.
-10
-5
5
10
-2
-4
-6
d.
x2  y1
y2  x1
6
B
4
A
C
D
Describe the transformation.
2
E
-5
5
-2
-4
-6
e.
x2  x1
y2  6  y1
6
B
4
Describe the transformation.
A
C
D
2
E
-5
5
-2
-4
-6
Tools of analytic geometry. Given two points (x1, y1 ) , (x2 ,y 2) :
Distance formula:
Midpoint formula:
Slope formula:
D  (x1  x2 )2  (y1  y2 )2
x  x 2 y1  y 2 
M   1
,

 2
2 
y y
m  2 1 Parallel lines have equal slopes.
x 2  x1
Perpendicular lines have opposite reciprocal slopes.
1. Here are the vertices of a triangle: A(2, -6), B(12, 18), C(-10, -1).
Calculate the slopes of the three sides.
mBC 
m AC 
m AB 
Why is the triangle a right triangle?
Which is the right angle?
Find the area of the triangle by any method you choose. Show your work.
2. Show that triangle PDQ is isosceles but not equilateral: P(1, 6), D(7, -1), Q(-1, -3).
3. A circle has a diameter AB with A( -1, 2) and B( 23, 9).
a. Find the center of the circle.
Center: (
b. Find the radius of the circle:
Radius:
,
)
Level 1 Geometry Review Answers
Review 7:
1. A regular polygon is a polygon with all sides congruent and all angles congruent.
2. 6840
3. 58
4. 4
5. 150
6. 720 sides
7. 36 sides
8. 42 sides
9. False
10. 72 sides 11. 24
12. 56.25, 56.25, 67.5
13. 34
14. 65
15. 50, 40
16. 9<x<19
Review 8:
1.
1. Givens 2. AF  CD opp. sides  3. AD opp s 4. AFBDCE ASA
5. AB  ED CPCTC 6. FD  AC opp sides  7. FE  BC Subtraction
8. FB  EC CPCTC 9. FBCE is a p-gram two pr of opp sides 
2.

3.
1. Givens 2. AR  BK Transitive 2. CARC Base Angles 3. ARCK
transitive 4. AR || BK Corr. Angles Thm 5. BARK is a p-gram one pr of opp
sides  and ||

1. Givens 2. NW  RT WT  NR opp sides of p-gram 
3. XW  RS VT  NP subtraction 4. WR NT opp s
5. XNPSTV WXVRSP SAS 6. XV  PS XP  VS CPCTC
7. XPSV is a p-gram opp sides of a p-gram 
4.
1. Givens 2. SO  RT RS  TO opp sides  3. RS || TO RT || SO opp sides ||
4. RSTSTO RTSTSO alt. int. s 5. RSMOTP MSOPTR
Supp  are  6. RSMOTP MSOPTR SAS 7. RM  PO RP  MO
CPCTC 8. MOPR is a p-gram opp sides of a p-gram are 
5.
6.
7.
8.
1.
1.
1.
1.
5.
9.
1. Givens 2. GH  GH Reflexive 3. FGHJHG SAS
4. FH  GJ JGH  FHG CPCTC 5. GK  KH CITT 6. FK  KJ
Subtraction 7. FKJ is isosceles Def. of isosceles
Givens 2. ABCACB Base Angles 3. 12 supp. of  are 
Givens 2. KP Base Angles 3. RKMRPO ASA 4. RM  RO CPCTC
Givens 2. SWXTZY SSS 3. WZ CPCTC 4. RW  RZ CITT
Givens 2. RS  RS Reflexive 3. PS  RT Addition 4. NPSVTR SAS
WRSWSR CPCTC 6. WR  WS CITT 7. WRS is isosceles Def of isosceles
Review 9:
1.
See me for solution
2.
1. Givens 2. BE  BD CITT 3. AEBCDB SAS 4. AC CPCTC
3.
1. Givens 2. DB  EC Subtraction 3. ABCACB ITT 4. BC  BC
Reflexive 5. DBCECB SAS 6. DCBEBC CPCTC 7. FB  FC
CITT 8. FBC is isosceles Def of isosceles.
Level 1 Geometry Review Answers
Review 4:
1.
a. SSS b. AAS c. AAS d. NN e. ASA or AAS f. AAS g. SAS h. NN i. HL
2.
1. Givens 2. CDYABX HL 3. YDC XBA CPCTC 4. AB||DC If alt.
int. s , lines are || 5. ABCD is a p-gram One pr of opp sides are  and ||
3.
a. 68 b. 57
4.
1. Givens 2. CAODBO If || lines, alt. int.  3. AOCDOB
vertical s 4. AOCBOD ASA
5.
a. 48 b. 64 c. 26 d. 20 e. 58 f. 80
Review 5:
1.
204.74 feet
2.
height = 14.82 area = 325.94
3.
x = 13, y = 51.98, z = 53.34
4.
0.979 miles
5.
Area of sector = 49.8, Area of triangle = 120.28, Area of segment = 49.8-120.8
6.
174.58
7.
688.19
2
1
8.
a. 1 b.
c.
2
2
Review 6:
1.
85
5.
32
8.
17.89 in
10.
x = 27, y = 4
2.
6.
9.
11.
33
3.
27.58
4.
They are all 78.5%
12.64
7.
base radius = 5, altitude = 15, lateral area = 150
volume = 7350, total surface area = 2688
x = 10, y = 5 12.
x = 27, y = 45
Level 1 Geometry Review Answers
Review 1:
1.
a.
b.
c.
d.
Venn Diagram
If you get a headache, then you banged your head against the wall.
Not necessarily true.
If you do not bang your head against the wall, you will not get a headache.
Not necessarily true.
If you do not get a headache, you did not bang your head against the wall.
True
Joe Max got a headache.
Ozzie has never banged his head against the wall.
No conclusion possible.
No conclusion possible.
Always true
neither
neither
Always true
e.
f.
g.
h.
2.
a.
b.
c.
d.
3.
113
4.
7.25
8 15
5.
65
 6.
7.
angles - 86, 66 arcs - 56, 132
8.
43
Review 2:
1.
front doors: 1:12; floor areas: 1:44; attic volumes: 1:1728
2.
18 sides
3.
120
4.
52.3%
Review 3:
Transformations:
1a.
translation 3 up and 5 to the left
1b.
reflection over the line y=-1
1c.
?
1d.
reflection over the line y= x
1e.
rotation about (0 , 3)
Coordinates:
12
19
5
m AB  ; m BC 
; m AC 
1.
5
22
12
The triangle is a right triangle because the slopes of 2 of the sides are opposite
reciprocals. A is the right angle.
Area of the triangle is 169 square units.
2.
PD = 49  36  85
DQ = 4  64  68
PQ = 4  81  85
Therefore, it is isosceles because PD = PQ.
3.
Center = (11 , 5.5)
Radius = 12.5