Complementary angles :
Two angles are complementary, if the sum of their measures is equal to 90.
Supplementary angles :
Two angles are supplementary angles if the sum of their measures is equal to 180 degrees.
Problem 1 :
Angles A and B are complementary. If m∠A = 3x - 8 and m∠B = 5x + 10, what is the measure of each angle ?
Solution :
Since angles A and B are complementary,
m∠A + m∠B = 90
3x - 8 + 5x + 10 = 90
8x + 2 = 90
Subtracting 2 on both sides.
8x = 90 - 2
8x = 88
Dividing by 8 on both sides.
x = 88/8
x = 11
m∠A = 3x - 8 = 3(11) - 8 = 33 - 8 m∠A = 25 |
m∠B = 5x + 10 = 5(11) + 10 = 55 + 10 m∠B = 65 |
So, the required angles measures are 25° and 65°.
Problem 2 :
Angles Q and R are supplementary. If m∠Q = 4x + 9 and m∠R = 8x + 3, what is the measure of each angle ?
Solution :
Since angles Q and R are supplementary, they add up to 180 degree.
m∠Q + m∠R = 180
4x + 9 + 8x + 3 = 180
12x + 12 = 180
Subtracting 12 on both sides.
12x = 180 - 12
12x = 168
Dividing by 12 on both sides, we get
x = 168/12
x = 14
m∠Q = 4x + 9 = 4(14) + 9 = 56 + 9 m∠Q = 65 |
m∠R= 8x + 3 = 8(14) + 3 = 112 + 3 m∠R = 115 |
So, the angle measures are 65 and 115.
Problem 3 :
Find the measure of two complementary angles ∠A and ∠B, if m∠A = 7x + 4 and m∠B = 4x+ 9.
Solution :
Since ∠A and ∠B are complementary, they add upto 90.
m∠A + m∠B = 90
7x + 4 + 4x + 9 = 90
11x + 4 + 9 = 90
11x + 13 = 90
Subtracting 13 on both sides.
11x = 90 - 13
11x = 77
x = 77/11
x = 7
m∠A = 7x + 4 = 7(7) + 4 = 49 + 4 m∠A = 53 |
m∠B = 4x + 9 = 4(7) + 9 = 28 + 9 m∠B = 37 |
So, the required angles are 53 and 37.
Problem 4 :
The measure of an angle is 44 more than the measure of its supplement. Find the measures of the angles.
Solution :
Let x be the required angle, its supplement be 180-x.
x = 180-x + 44
x = 224 - x
Add x on both sides.
x + x = 224
2x = 224
Divide by 2.
x = 224/2
x = 112
180 - x ==> 180 - 112 ==> 68
So, the required angles are 112 and 68.
Problem 5 :
What are the measures of two complementary angles if the difference in the measures of the two angles is 12.
Solution :
Let x be a angle, its complementary angle is 90 - x.
x - (90 - x) = 12
x - 90 + x = 12
2x = 12 + 90
2x = 102
Dividing by 2 on both sides.
x = 102/2
x = 51
90 - 51 ==> 39
So, the required angles are 39 and 51.
Problem 6 :
Find the measures of two supplementary angles ∠N and ∠M if the measure of angle N is 5 less than 4 times the measure of angle M.
Solution :
∠N = 4∠M - 5
∠N and ∠M are supplementary.
∠N + ∠M = 180
4∠M - 5 + ∠M = 180
5∠M = 180 + 5
5∠M = 185
Dividing by 5
∠M = 185/5
∠M = 37
Applying the value of ∠M, to find ∠N.
∠N = 4(37) - 5
∠N = 148 - 5
∠N = 143
Problem 7 :
Suppose ∠T and ∠U are complementary angles. Find x, if ∠T = 16x - 9 and ∠U = 4x - 1.
Solution :
Since ∠T and ∠U are complementary angles.
∠T + ∠U = 90
16x - 9 + 4x - 1 = 90
20x - 10 = 90
Add 10 on both sides.
20x = 90 + 10
20x = 100
x = 100/20
x = 5
∠T = 16(5) - 9 = 80 - 9 = 71 |
∠U = 4(5) - 1 = 20 - 1 = 19 |
So, the required angles are 71 and 19.
Problem 8 :
Two angles are vertical in relation. One angle is 2y and the other angle is y + 130. Find each angle measure.
Solution :
If two angles are vertical, then they will have the same measure.
2y = y + 130
2y - y = 130
y = 130
2y = 260 and y + 130 = 260
So, those two angels are 260 and 260.
Problem 9 :
The measure of two supplementary angles are in the ratio 4 : 2, Find those two angles.
Solution :
Let the required angles be 4x and 2x.
4x + 2x = 180
6x = 180
Divide by 6, we get
x = 180/6
x = 30
4x = 4(30) ==> 120
2x = 2(30) ==> 60
So, the required those two angles are 60 and 120.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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