Download Cat 2 - 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?
→
B
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
E
C
D
3. How many degrees are in an exterior angle of a regular 18-gon?
Answers
1. _____________
2. _____________
3. _____________
Solutions to Category 2
Geometry
Meet #1, October, 2000
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.
Answers
1. 105 degrees
2. 142 degrees
3. 20 degrees
B
A
E
C
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.°
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 .
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
D
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?
H
B
A
Answers
1. _____________
2. _____________
3. _____________
I
G
F
E
Solutions to Category 2
Geometry
Meet #1, October, 2001
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.
Answers
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.
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.
øæ
l
m
33°
33¼
68°
68¼
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?
C
G
Answers
1. _______________
2. _______________
3. _______________
D
B
A
F
E
Solutions to Category 2
Geometry
Meet #1, October, 2002
Answers
1. 192
2. 35
3. 117
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. 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.
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?
M
T
B
G
O
D
P
J
H
K
I
Q
m
n
R
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
less than 180 degrees?
S
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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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
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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?
k
l
.G
.
m
.
.D
.
E
.
.F
B
.A
C
n
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
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. _______________
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Solutions to Category 2
Geometry
Meet #1, October 2004
Answers
1. 120
2. 15
3. 67.5
60
60
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°.
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
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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.
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?
B
H
E
F
G
I
D
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?
Answers
1. _______________
2. _______________
3. _______________
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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
60
B
60
80
F
40
E 100
140
40
H
24
116
G
116
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.
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270 − 2x = 8x −10
+10
= + 10
280 − 2x = 8x
+2x = +2x
280 = 10x
x = 28
Category 2
Geometry
Meet #1, October 2006
1. In heptagon ABCDEFG, drawn accurately at right,
all angles are multiples of 45 degrees. How many
degrees are in the sum of interior angles D and E?
A
B
D
E
G
C
F
H
N
2. In the figure at left, angles HIJ, HKM, and
MLN are right angles. If the measure of angle
MNL is 36 degrees, how many degrees are in the
measure of angle HJI?
K
M
L
I
J
3. If you subtract twice an angle from its supplement, you get half its complement.
How many degrees are in the measure of this angle?
Answers
1. _______________
2. _______________
3. _______________
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Solutions to Category 2
Geometry
Meet #1, October 2006
Average score: 1.3 answers correct
90
Answers
1. The angle measures are given here.
The desired sum is 315 + 225 = 540.
1. 540
2. 54
H
N
M
315
45
225
90
3. 54
L
Some Incorrect
Answers Seen
1. 180
2.
3. 0, 60, 90
45
2. Triangles HIJ, HKM, and NLM are similar triangles.
This means they have the same angle measures. We
know that angle MLN is 90 degrees and angle MNL is
36 degrees, so angle LMN must be 180 – 90 – 36 = 54
degrees. Angle HJI is also 54 degrees.
K
I
90
J
3. If we call our unknown angle x, then twice the angle
is 2x, the supplement is 180 – x and the complement is
90 – x. Translating the sentence to algebra, we get the
following equation:
90 − x
(180 − x) − 2x =
2
Simplifying the left side, we get
90 − x
180 − 3x =
2
Doubling both sides, we get
360 − 6x = 90 − x
Adding 6x to both sides of the equation, we get
360 = 90 + 5x
Subtracting 90 from both sides, we get
270 = 5x
Finally, dividing both sides by 5, we find that 54 = x. So
the measure of the unknown angle is 54 degrees.
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