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Angles
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Angles: Problems with Solutions
By Catalin David
Problem 1
An angle measures:
the length of the lines
the opening between two lines
Solution:
The angle measures the opening between two lines that intersect. The lines are infinite, they don't affect the measure of the angle.
Problem 2
The unit of measurement used for angles is:
degrees
centimeters
kilograms
Solution:
Angles are measured using degrees.
Problem 3
Which point is the vertex of the angle?
B
C
A
Solution:
The vertex of the angle is the point shared by the two lines
In this case, point A.
Problem 4
If an angle is designated by only one letter, then that letter represents:
one of the sides of the angle
the vertex of the angle
Solution:
The letter that designates an angle represents the vertex of that angle.
Problem 5
If an angle is designated by 3 letters, then the one in the middle represents:
a point on one of the sides
the vertex of the angle
Solution:
If an angle is designated by 3 letters, then the middle one represents the vertex of the angle. The other two represent two points, each on one of the sides.
Problem 6
Are ∠BAC and ∠ABC the same angle?
Yes
No
Solution:
No, because the middle letter is not the same, so the angles don't have the same vertex.
Problem 7
Are ∠BAC and ∠DAE the same angle?
Yes
No
Solution:
Yes, because the middle letter is the same, so they have the same vertex. Moreover, points B and D are on one of the sides of the angle and points C and E are on the other side.
Problem 8
The measure of an acute angle is:
equal to 90°
greater than 90°
less than 90°
Solution:
Less than 90°
Problem 9
∠BAC is:
obtuse
right
acute
Solution:
∠BAC is acute because its measure is less than 90°.
Problem 10
The measure of a right angle is:
less than 90°
90°
greater than 90°
45°
Solution:
The measure of a right angle is 90°. Its sides are perpendicular.
Problem 11
An angle whose sides are perpendicular has a measure of:
90°
45°
180°
Solution:
90°
Problem 12
What type of angle is ∠ABC?
Right
Acute
Obtuse
Solution:
∠ABC is an acute angle because its measure is less than 90°. It can be included inside a right angle.
Problem 13
The measure of an obtuse angle is
less than 90°
greater than 90°
equal to 90°
Solution:
The measure of an obtuse angle is greater than 90°.
Problem 14
∠ABC is
right
acute
obtuse
Solution:
∠ABC is obtuse because its measure is greater than 90°.
Problem 15
∠ABC is
right
acute
obtuse
Solution:
∠ABC is obtuse because its measure is greater than 90°. A right angle can be included inside it.
Problem 16
An angle with both sides on the same line is
an acute angle
an obtuse angle
a straight angle
Solution:
A straight angle.
Problem 17
The measure of a straight angle is:
90°
180°
120°
0°
Solution:
The measure of a straight angle is 180° because two right angles can fit inside it.
Problem 18
А bisector is the half-line which splits an angle into
2
3
4
equal angles.
Solution:
The bisector splits the angle into 2 equal parts. In the figure, AC is the bisector of ∠BAD.
Problem 19
If the measure of an angle is 72°, the bisector forms two angles of
24°
36°
12°
each.
Solution:
The bisector forms two equal angles, each having a measure of 36°.
Problem 20
Are ∠BAC and ∠DEF adjacent?
No
Yes
Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, these requirements are not met, so the angles are not adjacent.
Problem 21
Are ∠BAC and ∠DAE adjacent?
Yes
No
Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, the angles only share the same vertex, so they are not adjacent.
Problem 22
Are ∠BAC and ∠EDC adjacent?
No
Yes
Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, the angles only share one of their sides, so they are not adjacent.
Problem 23
Are ∠BAC and ∠CAD adjacent?
Yes
No
Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case, all requirements are met, so the angles are adjacent.
Problem 24
Are ∠BAC and ∠BAD adjacent?
No
Yes
Solution:
Two angles are adjacent if they share the same vertex and one of their sides and if they don't share any interior points. In this case the angles share interior points, so they are not adjacent.
Problem 25
∠ABC has a measure of 42° and ∠CBD has a measure of 36°. Is the measure of ∠ABD 78°?
Yes
No
Solution:
Yes, since ∠ABC and ∠CBD are adjacent angles.
Thus, ∠ABD = ∠ABC + ∠CBD = 42° + 36° = 78°.
Problem 26
∠ABC has a measure of 56° and ∠ABD has a measure of 100°. Is the measure of ∠DBC 44°?
Yes
No
Solution:
Yes, since ∠ABC and ∠CBD are adjacent angles. Thus, ∠DBC=∠ABD-∠ABC=100°-56°=44°.
Problem 27
Two angles are complementary if the sum of their measures is
180°
90°
45°
60°
Solution:
Two angles are complementary if the sum of their measures is 90°.
Problem 28
Are ∠A and ∠B complementary?
Yes
No
Solution:
The sum of their measures is 43° + 47° = 90°, so the angles are complementary.
Problem 29
If OA and OC are perpendicular, are ∠AOB and ∠BOC complementary?
Yes
No
Solution:
If OA and OC are perpendicular, then ∠AOC has a measure of 90°. ∠AOB and ∠BOC are adjacent. ∠AOC=∠AOB+∠BOC=90°. The angles are complementary.
Problem 30
The measure of ∠AOB is
24°
66°
80°
76°
Solution:
∠AOB and ∠BOC are adjacent and complementary.
In this case, ∠AOB + ∠BOC = 90°.
∠AOB + 24° = 90°.
∠AOB = 90° - 24° = 66°
Problem 31
If an angle has a measure of 40°, the measure of its complementary is
40°
80°
50°
.
Solution:
The sum of the measures of complementary angles is 90°. If one of the angles has a measure of 40°, the measure of the other one is 90°-40°=50°.
Problem 32
Two angles are supplementary if the sum of their measures is
180°
90°
60°
45°
Solution:
The sum of the measures of supplementary angles is 180°.
Problem 33
Are ∠AOB and ∠BOC supplementary?
Yes
No
It is undefined
Solution:
∠AOB and ∠BOC are adjacent. Thus, ∠AOB+∠BOC=∠AOC. But ∠AOC's sides are on the same line, so its measure is 180°. Hence, ∠AOB and ∠BOC are supplementary.
Problem 34
What is the measure of ∠BOC?
90°
135°
45°
145°
Solution:
∠AOB and ∠BOC are adjacent and supplementary. Thus m∠AOB + m∠BOC = 180°.
45° + m∠BOC = 180°.
m∠BOC = 180° - 45° = 135°.
Problem 35
If ∠AOB and ∠COD are complementary, what is the measure of ∠BOC?
60°
90°
120°
80°
Solution:
The angles are adjacent two by two. Thus, ∠AOB+∠BOC+∠COD=∠AOD. But ∠AOD's sides are on the same line, so its measure is 180°. ∠AOB and ∠COD are complementary, so ∠AOB+ ∠COD=90°. ∠BOC + 90° = 180°, so ∠BOC = 90°.
Problem 36
∠AOB and ∠BOC are complementary. ∠BOC and ∠COD are complementary. Are ∠AOB and ∠COD equal?
Yes
No
Solution:
If ∠AOB and ∠BOC are complementary, then ∠AOB+∠BOC=90°. If ∠BOC and ∠COD are complementary, then ∠BOC+∠COD=90°.
∠AOB+∠BOC=∠BOC+∠COD. Thus, ∠AOB=∠COD, so two angles having the same complementary angle are equal.
Problem 37
∠AOB and ∠BOC are complementary. ∠BOC and ∠COD are complementary. If ∠BOC has 30°, what is the measure of ∠AOD?
120°
150°
180°
Solution:
If ∠AOB and ∠BOC are complementary, then ∠AOB+∠BOC=90°.
If ∠BOC and ∠COD are complementary, then ∠BOC+∠COD=90°.
∠AOB+∠BOC=∠BOC+∠COD.
Thus, ∠AOB=∠COD=60°,
∠AOD=∠AOC+∠COD=90°+60°=150°.
Problem 38
Lines AC and BD meet in point O. Are ∠AOB and ∠COD opposite angles?
Yes
No
Solution:
Yes, since opposite angles are formed by two intersecting lines.
Problem 39
Are ∠AOB and ∠COD opposite angles?
Yes
No
Solution:
No, since opposite angles are formed by two intersecting lines. Points A, O and C are not on the same line. Points B, O and D are not on the same line either.
Problem 40
∠AOB and ∠COD are opposite angles. Do they have the same measure?
No
Yes
Solution:
∠AOB and ∠DOA are adjacent.
∠AOB+∠AOD=∠BOD=180°.
∠COD and ∠DOA are adjacent.
∠COD+∠DOA=∠COA=180°.
Thus, ∠AOB and ∠COD have the same supplementary and are equal.
Problem 41
∠AOB and ∠COD are opposite angles. If ∠AOB=35°, what is the measure of ∠COD?
15°
66°
150°
35°
Solution:
Since they are opposite angles, they have the same measure. ∠COD has a measure of 35°.
Problem 42
∠AOB and ∠COD are opposite angles. If ∠AOB = 35°, what is the measure of ∠DOA?
150°
145°
90°
135°
Solution:
∠DOA and ∠AOB are supplementary. In this case, m∠AOB + m∠DOA = 180°, so ∠DOA = 145°.
Problem 43
Let a and b be two lines intersected by line c. Find two alternate interior angles.
3 and 4
4 and 5
3 and 5
Solution:
Alternate interior angles are non-adjacent, inside the space formed by lines a and b, on one side and the other of line c (the secant line). Two angles that match these properties are 3 and 5, also 4 and 6.
Problem 44
Let a and b be two lines intersected by line c. Find two alternate exterior angles.
2 and 8
1 and 3
2 and 5
Solution:
Alternate exterior angles are non-adjacent, outside the space formed by lines a and b, on one side and the other of line c (the secant line). Two angles that match these properties are 2 and 8, also 1 and 7.
Problem 45
Let a and b be two lines intersected by line c. Find two corresponding angles.
3 and 8
2 and 7
3 and 7
1 and 6
Solution:
Corresponding angles are non-adjacent, one of them outside the space formed by lines a and b and the other inside, both on the same side of line c (the secant line). Two angles that match these properties are 3 and 7.
Other pairs are 2 and 6, 1 and 5, 4 and 8.
Problem 46
Are lines a and b parallel?
No
Yes
Solution:
The two angles are alternate exterior angles. They have the same measure, so lines a and b are parallel.
Problem 47
Are lines a and b parallel?
Yes
No
Solution:
The two angles are alternate interior angles, but they have different measures. In this case, lines a and b are not parallel.
Problem 48
Let a and b be two parallel lines intersected by line c. If m∠1 = 75°, what is the measure of ∠7?
105°
15°
75°
95°
Solution:
The two angles are alternate exterior angles. Since lines a and b are parallel, the angles will have the same measure, so m∠7 = 75°.
Problem 49
Let
a
and
b
be two parallel lines intersected by line
c
. If m∠2 = 120°, what is the measure of ∠6?
60°
120°
150°
100°
Solution:
The two angles are corresponding angles. Since lines a and b are parallel, the angles will have the same measure, so ∠6 also has 120°.
Problem 50
Let a and b be two parallel lines intersected by line c. If ∠2 has 35°, what is the measure of ∠7?
35°
145°
120°
135°
Solution:
Since lines a and b are parallel, ∠2 and ∠8 have the same measure, so ∠8 also has 35°. However, ∠7 and ∠8 are adjacent supplementary angles, so the measure of
∠7 is 180° - 35° = 145°.
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