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Carlisle Math Team
Meet #1 – Category 2
M1C2
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Self-study Packet
Problem Categories for this Meet:
1. Mystery: ?
2. Geometry: Angle measures in plane figures including
supplements and complements
3. Number Theory: Divisibility rules, factors, primes, composites
4. Arithmetic: Order of operations; mean, median, mode; rounding;
statistics
5. Algebra: Simplifying and evaluating expressions; solving
equations with 1 unknown including identities
For current schedule or information,
see http://www.imlem.org
M1C1
GEOMETRY
ANGLES
To name an angle: use three points, one from each ray and the vertex.
Vertex letter is always in the middle.
Type of Angle
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
Reflex Angle
Description
an angle that is less than 90°
an angle that is 90° exactly
an angle that is greater than 90° but less
than 180°
an angle that is 180° exactly
an angle that is greater than 180°
Supplementary Angles
Two Angles are Supplementary if they add up to 180 degrees.
These two angles (140° and 40°)
are Supplementary Angles,
beacuse they add up to 180°.
Notice that together they make a
straight angle.
But the angles don't have to be
together.
These two are supplementary
because 60° + 120° = 180°
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If the two angles add to 180°, we say they "Supplement" each other.
GEOMETRY
Complementary Angles
Two Angles are Complementary if they add up to 90 degrees (a
Right Angle).
These two angles (40° and 50°) are
Complementary Angles, beacuse
they add up to 90°.
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ANGLE MEASURES IN PLANE FIGURES
In this example, a° and b° are vertical angles.
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a° = b°
vertical angles are equal: (in fact they are congruent angles)
Because b° is vertically opposite 40°, it must also be 40°
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GEOMETRY
A full circle is 360°, so that leaves 360° - 2×40° = 280°
Angles a° and c° are also vertical angles (and must be equal), which
means they are 140° each.
Answer: a = 140°, b = 40° and c = 140°.
A Transversal is a line that crosses at least two other lines.
Transversal
crossing two
lines
Transversal crosses
two parallel lines
transversal cuts
across three lines
Alternate Interior Angles.
the pairs of angles on opposite sides of the
transversal but inside the two lines
c and f
d and e
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GEOMETRY
Corresponding angles
the angles in matching corners
a an d e
b and f
c and g
d and h
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aa
Alternate Exterior Angles
the pairs of angles on opposite sides of the transversal but outside the
two lines
a and h
b and g
If any pair of:
Corresponding Angels are equal, or
Alternate Interior Angles are
equal, or
Alternate Exterior Angles are
equal, or
Consecutive Interior Angles
add up to 180°
a=e
c=f
b=g
d + f = 180°
... then the lines are
Parallel
M1C1
GEOMETRY
Interior Angles of Polygons
Triangles
The Interior Angles of a Triangle add up TO 180°
90° + 60° + 30° = 180°
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Quadrilaterals (Squares, etc)
The Interior Angles of a Quadrilateral add up to 360°
80° + 100° + 90° + 90° = 360°
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Pentagon
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made
A pentagon has 5 sides, and can be
from three triangles,
GEOMETRY
so you know
what...
... its internal angles add up to 3 × 180° = 540°
And if it is a regular pentagon (all angles the
angle is 540° / 5 = 108°
same), then each
(Exercise: make sure each triangle here adds up to 180°, and check that
the pentagon's internal angles add up to 540°)
The General Rule
So, each time we add a side (triangle to quadrilateral, quadrilateral to
pentagon, etc), we add another 180° to the total:
Shape
Sides
Sum of
Internal Angles
3
180°
Shape
Each Angle
Triangle
Q ui ck Ti m e ™ an d a
T IF F ( Un co m p e
r ss ed ) d ec om pr es s or
a re ne ed ed t o s ee th i s pi c u
t r e.
60°
Quadrilateral
4
360°
5
540°
6
720°
...
...
Q ui ck Ti m e ™ an d a
T IF F ( Un co m p e
r ss ed ) d ec om pr es s or
a re ne ed ed t o s ee th i s pi c u
t r e.
90°
Pentagon
Q ui ck Ti m e ™ an d a
T IF F ( Un co m p e
r ss ed ) d ec om pr es s or
a re ne ed ed t o s ee th i s pi c u
t r e.
108°
Hexagon
...
Q ui ck Ti m e ™ an d a
T IF F ( Un co m p e
r ss ed ) d ec om pr es s or
a re ne ed ed t o s ee th i s pi c u
t r e.
...
Any Polygon
...
(n-2) × 180° / n
n
M1C1
120°
(n-2) × 180°
Q ui ck Ti m e ™ an d a
T IF F ( Un co m p re ss ed ) d ec om pr es s or
a re ne ed ed t o s ee th i s pi c tu r e.
GEOMETRY
Category 2
Geometry
Meet #1, October 2005
1. Tim added x degrees to a 27-degree angle. The complement of this new angle was
48 degrees. He then added y degrees to this 48-degree angle. The complement of this
new angle was 7 degrees. Find the value of x + y.
Hint: What is the Complement of 48° ? Of 7°?
C
A
2. In the figure at right, the measure of
angle ABC is 60 degrees, the measure
of angle DEF is 100 degrees, and the
measure of angle DGI is 116 degrees.
How many degrees are in the measure
of angle FHG?
Hint: Write in the angles above, and then write in as
B
H
F
E
G
I
D
many angles as you can deduce.
3. The sum of the complement of angle x and the supplement of angle x is 10 degrees
less than eight times the angle x. How many degrees are in the measure of angle x?
Hint: Write an equation and solve it. (90 – x) + (180 – x) = …
Answers
1. _______________
2. _______________
3. _______________
M1C1
GEOMETRY
Solutions to Category 2
Geometry
Meet #1, October 2005
Answers
1. 50
2. 24
3. 28
1. The measures of two complementary angles add up to 90
degrees. Since 27 + 48 = 75, the unknown amount x must
have been 90 – 75 = 15 degrees. Likewise, since 48 + 7 =
55, y must be 90 – 55 = 35. The value of x + y is thus 15 +
35 = 50.
2. Vertical angles are congruent, straight angles have a sum
of 180 degrees, and triangles have an angle sum of 180
degrees. Using these three facts, the angles of the two small
triangular regions can be determined from the angle
measures given. The measure of angle FHG is 24 degrees.
C
A
B
F
H
E
G
I
D
3. The complement of angle x is 90 – x and the
supplement of angle x is 180 – x. Their sum is (90
– x) + (180 – x) = 270 – 2x. We know that this
sum is equal to ten less than eight times angle x, or
8x – 10. Now we can write an equation and solve
for x.
270  2x  8x 10
10
  10
280  2x  8x
2x  2x
280  10 x
x  28

M1C1
GEOMETRY
Category 2
Geometry
Meet #1, October 2004
1. Line l and line m are parallel.
The measure of angle DBA is 150°, and
the measure of angle CEF is 30°.
How many degrees are in the measure of
angle GAB?
Hint:CEF is the same as AGB. What is GBA? Now you
k
l
.G
.
m
.
.D
.
E
.
.
F
B
.A
C
n
can find GAB.
2. The eight-pointed star in the figure at
right was created by placing equilateral
triangles, such as A, along the inside edges
of a regular octagon. How many degrees
are in the angle measure of a point on the
star?
A
Hint: What is the exterior angle of an Octagon? 360/8.
Now what is the interior angle of an octagon? What is the
interior angle of an equilateral triangle?
3. If the supplement of angle x is five times the complement of angle x, how many
degrees are in the measure of angle x? Give your answer to the nearest tenth of a
degree.
Answers
1. _______________
2. _______________
3. _______________
M1C1
Hint: Make an equation and solve.
GEOMETRY
Solutions to Category 2
Geometry
Meet #1, October 2004
Answers
1. 120
2. 15
1. Angle GBA is supplementary to angle DBA, whose
measure is 150°, so the measure of angle GBA is 180° – 150°
= 30°. Angle BGA and angle CEF are corresponding angles,
so the measure of angle BGA must be equal to that of CEF,
which is also 30°. Triangle GAB must have a total of 180°,
so the measure of angle GAB is 180° – 2  30° = 120°.
3. 67.5
135°
Interior
angle
2. The interior angle of the regular octagon can be found in
several ways. One way is to subdivide the octagon into six
triangles, each of which has an angle sum of 180 degrees.
The total interior angle is thus 6  180 = 1080 degrees. In a
regular octagon, this total is shared equally among the eight
interior angles, so each of them has an angle of 1080 ÷ 8 =
135 degrees. Each equilateral triangle has three 60 degree
angles. The vertices of two triangles meet at each vertex of
the octagon, occupying 2  60 = 120 degrees of that angle.
The angle measure of a point on the star is the rest of the
interior angle, or 135 – 120 = 15 degrees.
3. The supplement of angle x is 180 – x, and the complement
of angle x is 90 – x. Translating the statement to algebra, we get 180  x  590  x.
Solving for x, we get
180  x  450  5x
180  x  5x  450  5x  5x 
180  4 x  450
180 180  4 x  450 180
22.5
4 x  270
112.5
4 x 270
67.5

4
4
x  67.5

M1C1
GEOMETRY
Category 2
Geometry
Meet #1, October 2003
1. Lines TP, BG, and DM intersect
at point O. mBOT  47 degrees
and mMOG  29 degrees.
How many degrees are in the measure

of angle DOP?

Hint: Find vertical angle to BOT. Add to MOG.

Find supplement.
B
K
I
m
Q
n
R
S
G
O
P
D
J
H
M
T
2. Lines m and n are parallel.
mHIJ 148 degrees and
mQRS 133 degrees.
How many degrees are in the
measure of angle IJK if it is
 degrees?
less than 180
 angles on the diagram.
Hint: Write known
Find angle JIK. Find IKR.
3. The sum of the supplement of angle A and the complement of angle A measures
sixteen degrees more than a straight angle. How many degrees are in the measure of
angle A?
Answers
1. _______________
2. _______________
3. _______________
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Hint: 180-A = Supplement. What is the
Complement? Write equation & solve.
GEOMETRY
Solutions to Category 2
Geometry
Meet #1, October 2003
Answers
1. 104
2. 101
1. The measures of angles BOT, TOM, and MOG must add
up to 180 degrees since O is a point on line BG. Thus the
measure of angle TOM must be 180 – 47 – 29 = 104 degrees.
Angles TOM and DOP are verticle angles and therefore have
the same measure. The measure of angle DOP is 104
degrees.
3. 37
2. Angle HIJ measures 148 degrees, so angle JIK must
measure 180 – 148 = 32 degrees. Angle QRS and angle IKR
are corresponding angles, so they have the same measure.
This means angle JKI must measure 180 – 133 = 47. The
total angle sum of triangle IJK has to be 180 degrees, so
angle IJK must measure 180 – 32 – 47 = 101 degrees.
3. The supplement of angle A measures 180 – A. The
complement of angle A measures 90 – A. Their sum is
180  A  90  A  270  2A . If this amount is sixteen
degrees more than a straight angle, then we can write the
equation 270  2A 180 16 and solve for A.

270  2A  180  16
270  2A  196

270  196  2A
270 196  2A
74  2A
A  37

M1C1
GEOMETRY
Category 2
Geometry
Meet #1, October, 2002
1. The measure of a certain angle a is 39 degrees. Let s be the supplement of this
angle a and let c be the complement of angle a. How many degrees are there in the
measure of angle (s + c)?
2. Line l is parallel to line m. Find the measure of angle ø in degrees.
ø
33°
68°
3. In the figure shown at right, angles CAE, GFE, and CDB are right angles and angle
ACE measures 27 degrees. How many degrees are in the measure of the angle DGF?
Answers
1. _______________
2. _______________
3. _______________
M1C1
Hint: Knowing the angle at C, write the angles at
B. Now you can find the angles at G.
GEOMETRY
Solutions to Category 2
Geometry
Meet #1, October, 2002
Answers
1. 192
1. If s is the supplement of a 39 degree angle, then
s  180 39  141. If c is the complement of angle a, then
c  90 39  51. Thus the value of s + c is 141 + 51 = 192.
2. 35
3. 117
2. The obtuse angle in the triangle is the supplement of 68
degrees, or 180 – 68 = 112 degrees. Since every triangle has
a total of 180 degrees, ø must be 180 – 112 – 33 = 35
degrees. A shorter way to arrive at this result is 68 – 33 =
35, since an exterior angle of a triangle is equal to the sum of
the two non-adjacent angles.
3. If the measure of angle ACE is 27 degrees, then the
measure of angle AEC must be 90 – 27 = 63 degrees. The
angle sum in quadrilateral DEFG must be 360 degrees.
Angles EFG and EDG are right angles and AEC is 63
degrees. Thus the measure of angle DGF is 360 – 90 – 90 –
63 = 117 degrees.
Another approach: Exterior angle ABD must equal the sum
of the two interior angles ACE =27° and right angle CDB, so
ADB=117°. AB and FG are parallel (since both are
perpendicular to the same line) and so GBA=DGF=117°.
M1C1
GEOMETRY
Category 2
Geometry
Meet #1, October, 2001
1. In the figure to the right, angles GRM and
MRT are complementary. Angles MRT and
TRY are supplementary. The measure of angle
TRY is 127 degrees. How many degrees are in
the measure of angle GRY?
G
M
R
Y
T
A
2. In the figure to the left, angles ABC and ADE are
right angles. FED and FBC are straight lines. The
measure of angle ACB is 55 degrees. How many
degrees are in the measure of angle EFB?
D
E
F
B
C
3. In the figure to the right, regular pentagon
AGHIF sits inside regular hexagon ABCDEF so that
the two shapes share base AF. How many degrees
are in the measure of angle GAB?
D
C
H
B
I
G
E
Hint: Find the exterior angle for each shape and subtract.
A
F
Answers
1. _____________
2. _____________
3. _____________
M1C1
GEOMETRY
Solutions to Category 2
Geometry
Meet #1, October, 2001
Answers
1. We know that angle TRY measures 127, so its
supplement, angle MRT, must be 180  127  53
degrees. Angle GRM, the complement to angle MRT,
must be 90  53  37 degrees. Angle GRY is the
supplement to GRM, so it must be 180  37  143
degrees. Alternatively, we might notice that angles GRY
and TRY must have a sum of 270 degrees (GRM and
MRT together make the other 90 degrees in the 360
degrees around point R) and 270  127  143 degrees.
1. 143
2. 35
3. 12
G
37
M
143
R
Y
53
127
T
A
35
55
E
55
35
F
90
D
90
125
90 90
B
55
C
2. The figure contains four similar right triangles:
ABC, ADE, FBE, and FDC. All four of these
triangles have an angle that measures 90 degrees
and an angle that measures 55 degrees. The third
angle must equal 35 degrees since the total angle
sum of any triangle is 180 degrees and
90  55  35  180 . In particular, triangle FDC
contains a right angle at D and a 55 degree angle
at C, so the angle at F must be 35 degrees.
3. Regular hexagons have interior angles of 120 degrees and regular pentagons have
interior angles of 108 degrees. This can be determined by partitioning the polygon into
triangles, each containing 180 degrees. Thus, the measure of angle GAB is
120  108  12 degrees.
M1C1
GEOMETRY
Category 2
Geometry
Meet #1, October, 2000
1. A regular octagon and a regular hexagon share a common side. What is the number of degrees in
the measure of the exterior angle formed where they meet?


2. In the figure, rays AB and DE are parallel.
Angle C is a right angle and angle B measures 52
degrees. Find the measure of angle D if it is less
than 180 degrees.
A
B
E
C
D
3. How many degrees are in an exterior angle of a regular 18-gon?
Answers
1. _____________
2. _____________
3. _____________
M1C1
GEOMETRY
Solutions to Category 2
Geometry
Meet #1, October, 2000
Answers
1. The interior angle measures of the regular octagon and
the regular hexagon are 135 degrees and 120 degrees
respectively. 135  120  255 , so the exterior angle
between the shapes must account for the remaining 105
degrees of a full circle.
1. 105 degrees
2. 142 degrees
3. 20 degrees
2. By extending ray AB and creating a line parallel to
segment BC, we can see that angle D is composed of a 90
degree angle and the same 52 degrees that is found at B.
Thus, angle D is 142 degrees.
B
A
E
C
D
3. A regular 18-gon can be divided into 16 triangles each
with an angle sum of 180 degrees. The total angle sum of
the 18-gon is 16  180  2880 . Since the 18-gon is
regular, each interior angle is one eighteenth of 2880, or
160 degrees. An exterior angle is the supplement of the
interior angle, which is 20 degrees in this case.
Alternatively, some students will know that the sum of all
the exterior angles of a polygon is always 360. Since the
18-gon is regular, we simply divide as follows:
360  18  20 .
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GEOMETRY
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GEOMETRY
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GEOMETRY