Geometry in a Nutshell
I.
Basic Angle Relationships
Key Terms and Facts



The number of degrees in a straight angle is 180.
The sum of the angles in a triangle must be 180 degrees.
The exterior angle of a triangle is equal to the sum of the two remote interior
angles of the same triangle (see diagram). Here, a  m  p .
Define the following:
Vertical Angles:
Adjacent Angles:
Supplementary Angles:
Complementary Angles:
II.
Angles Involving Parallel Lines
Key Terms and Facts

See diagram
Define the following:
Alternate Interior Angles:
Alternate Exterior Angles:
Corresponding Angles:
Consecutive Interior (Same-Side Interior) Angles:
III.
Pythagorean Theorem
Recall that:
a2  b2  c2
In other words, the sum of the squares of the legs of a right triangle is equal to the square
of the hypotenuse.
It also might be helpful to memorize the following Pythagorean Triples, as they do
appear regularly:
3, 4, 5
5, 12, 13
6, 8, 10
7, 24, 25
9, 40, 41
IV.
Special Right Triangles
Refer to the following diagram to aid you in completing problems involving special right
triangles.
45
30
2x
x 2
x
x 3
45
x
60
x
Warm-up #1
1)
What is the value of x?
2)
If x  2 y , what is the value of y?
Warm-up #2
Find all of the angles in the figure below. Take special note of the information provided
to you under the diagram.
Z
XY TS
SZ bisects YST
Warm-up #3
Latisha has a circular swimming pool with a radius of 8 feet. Jason told Latisha that if
the radius of the pool were twice as long, or 16 feet, she would have twice the area to
swim around in. Latisha insisted that Jason was wrong and that she would have four
times as much area to swim in. Who do you agree with? Justify your answer.
Problem Set #1
1)
In the figure above, n 
(a)
(b)
(c)
(d)
(e)
3)
In the figure below, what is the
value of m?
60
65
70
75
80
Note: Figure not drawn to scale.
__________
4)
In the triangle below, what is
the value of a?
Note: Figure not drawn to scale.
2)
What is the value of x in the
figure above?
(a)
(b)
(c)
(d)
(e)
30
40
45
50
60
__________
Problem Set #2
a
3)
1)
In the figure above, five lines
intersect as shown. If lines l, m,
and n are parallel, what is the
value of a + b?
(a)
(b)
(c)
(d)
(e)
210
220
230
240
250
Note:
Figure not drawn to scale.
If x is parallel to y in the figure
above, what is the value of a?
(a)
(b)
(c)
(d)
(e)
30
45
60
120
135
In the figure above, line a is
parallel to line b, and segment
WY bisects angle XWZ. What is
the value of n?
_____________
4)
2)
b
If lines a, b, and c are parallel
and lines d and e intersect at
point P, which lies on line a, how
many degrees is the measure of
angle x?
Problem Set #3
1)
In the figure to the right, what is the perimeter of triangle ABC?
(a)
10  34
(b)
15  3
15  3
5  10
5  15
(c)
(d)
(e)
2)
3
C
A
6
4
PR is tangent to a circle with center O and radius 4 at point Q. If the measure of
20
, then PQ =
POR is 90  , OR = 5, and OP =
3
(a)
(b)
(c)
(d)
(e)
3)
3
5
5
3
B

7
3
7
3
16
3
20
3
23
3
If the area of the triangle shown at right is 100, what is the length of side AB?
B
(a)
10 3
(b)
(c)
(d)
(e)
10 5
20
24
25
20
A
C
Note: Figu re not drawn to scal e.
4)
In the figure below, SU is tangent to the circle with center R at point T. If
ST has a length of 40 and the area of the circle is 81 , what is the length of
SR ?
(a)
(b)
(c)
(d)
(e)
31
35
41
45
Cannot be determined from the
information given
T
S
U
R
Note: Figu re not drawn to scale .
Problem Set #4
1)
Diameter AC is 9 2 units long and is also the diagonal of square ABCD
whose vertices are points on circle O.
What is the perimeter of square ABCD?
D
C
(a) 9 2
(b) 18
(c) 36
O
(d) 72
(e) 81
B
A
2)
A square, KLMN, is inscribed inside another square that has a side of length
2 2 . What is the area of KLMN ?
K
(a)
(b)
(c)
(d)
(e)
2 2
4
4 2
22 2
5
L
N
M
3)
In the triangle at right, DC=BC, AC is 6 3 . What is the area of triangle
BCD?
A
30
D
Answer: _______________
B
C E F
G
4)
Paul is building a gingerbread house. The front face of the house is formed by
a square and a right triangle, as shown in the figure above. If the height of the
square is 10 inches, then what is the combined length, in inches of the two
sides of the triangle labeled x?
(a) 10
(b) 15
10
(c)
2
20
(d)
2
30
(e)
3
5)
x
x
Note: Figu re not drawn to scal e.
In the rectangle ABCD to the right, if ABE is equilateral, and BC = 9, what
is the sum of the lengths of AE and BE ?
C
B
(a) 9
(b) 18
(c) 12 3
(d) 18 3
(e) 36
E
D
A
Note: Figure not drawn to scal e.
Problem Set #5
1)
In the figure above, inscribed polygon
ABCDEF is equilateral. If the diameter
circle is 12, then the length of arc BCD is
(a)
(b)
(c)
(d)
(e)
2)
of the
12
8
6
4
2
In the figure above, triangle PQR has
sides of lengths x, y, and 0.5(x+y). On
each
side, a square is constructed as
shown. What is the sum of the lengths
of the sides of the
resulting 9-sided figure, in terms of x and y?
(a)
(b)
(c)
(d)
(e)
9x  9 y
2
7x  7 y
2
3x  3 y
2
5x  5 y
4x  4 y
3)
Three circles A, B, and C exist with the following conditions: the radius of A is half the radius of
B, and the radius of B is half the radius of C. If the radius of B is 4, what is the difference in the
areas of A and C?
(a)
(b)
(c)
(d)
(e)
4)
4
6
16
32
60
If the perimeter of a rectangle is eight times the length, then the width of the rectangle is how
many times the length?
And if you’re up for a CHALLENGE…..
1)
In the figure above, A and B are the centers of the two circles. If each circle has area 10, what is
the area of the rectangle?
(a)
(b)
20
(c)
40
(d)
50
(e)
60
20 



10

r
2)
Which of the following is equal to the perimeter of the figure above?
(a)
(b)
(c)
(d)
(e)
r  s a b
2r  s  ( a  b)
2(r  s )  (a  b)
2(r  s )  (a  b)
2(r  s)
Note: Figure not drawn to scale.
3)
The figure above is composed of two semi-circles and one triangle. What is the perimeter of the
figure?
(a)
(b)
(c)
(d)
(e)
6  10
7  7
7  10
14  7
14  10
4)
The half-circle shown above has a radius of 6 and the center is D. If
of the unshaded region?
(a)
(b)
(c)
(d)
(e)
5)
AC  BD , what is the area
18  9
18 18
18  6 3
36  9
36 18
In a square with vertices WXYZ, if point V is the midpoint of side YZ and the area of triangle
XYV is 4 , what is the area of square WXYZ?
5
(a)
(b)
2
(c)
(d)
4
(e)
6)
8
5
16
5
18
5
An equilateral triangle is inscribed in a circle with radius 2. What is the area of the
triangle?
(a) 3 3
(b) 2 3
(c) 3 2
(d)  6
(e)  10
7)
The area of the square in the diagram is equal to s2. What fraction of its area is
occupied by the four circles?
(a)
8)
s 2
4
(b)
6s 2
3
(c)
2
5
(d)
6
6
(e)

4
Circles L and R have radius 4. The radius of circle M is 2 and line LMR is collinear
with the diameter of the largest circle. What is the area of the shaded region?
(a) 64
(b) 48
(c) 42
(d) 32
(e) 24