Problem 1 :
Find the measure of an angle that is twice as large as its supplement.
Solution :
Let x be the measure of the required angle.
The measure of a supplement of the angle x is (180° - x).
Given : The measure of an angle that is twice as large as its supplement.
x = 2(180° - x)
x = 360° - 2x
Add 2x to both sides.
3x = 360°
Divide both sides by 3.
x = 120°
Therefore, the required angle is 120°.
Problem 2 :
Find the measure of an angle that is half as large as its complement.
Solution :
Let x be the measure of the required angle.
The measure of a complement of the angle x is (90° - x).
Given : The measure of an angle that is half as large as its complement.
Multiply both sides by 2.
2x = 90° - x
Add x to both sides.
3x = 90°
Divide both sides by 3.
x = 30°
Therefore, the required angle is 30°.
Problem 3 :
The measure of a supplement of an angle is 12 more than twice the measure of the angle. Find the measures of the angle and its supplement.
Solution :
Let x be the measure of the required angle.
The measure of a supplement of the angle x is (180° - x).
Given : The measure of a supplement of an angle is 12 more than twice the measure of the angle.
180° - x = 2x + 12°
Add x to both sides.
180° = 3x + 12°
Subtract 12 from both sides.
168° = 3x
Divide both sides by 3.
56° = x
And also,
180° - x = 180° - 56° = 124°
Therefore, the measure of the angle is 56° and its supplement is 124°.
Problem 4 :
A supplement of an angle is six times as large as a complement of the angle. Find the measures of the angle, its supplement and its compliment.
Solution :
Let x be the measure of the required angle.
The measure of a complement and a supplement of the angle x are (90° - x) and (180° - x) respectively.
Given : A supplement of an angle is six times as large as a complement of the angle.
180° - x = 6(90° - x)
180° - x = 540° - 6x
Add 6x to both sides.
180° + 5x = 540°
Subtract 180° from both sides.
5x = 360°
Divide both sides by 5.
x = 72°
Therefore, the measure of the angle is 72°, its supplement is 108° and its complement is 18°.
Problem 5 :
You are told that the measure of an acute angle is equal to the difference between the measure of a supplement of the angle and twice the measure of a complement of the angle. What can you deduce about the angle? Explain.
Solution :
Let x be the measure of an acute angle.
The measure of a complement and a supplement of the angle x are (90° - x) and (180° - x) respectively.
Given : The measure of an acute angle is equal to the difference between the measure of a supplement of the angle and twice the measure of a complement of the angle.
x = (180° - x) - 2(90° - x)
x = 180° - x - 180° + 2x
x = x
The above is equation is true for all values of x such that x is an acute angle.
Conclusion :
The measure of any acute angle is equal to the difference between the measure of a supplement and twice the measure of a complement of the acute angle.
Problem 6 :
Can the measure of a complement of an angle ever equal exactly half the measure of a supplement of the angle? Explain.
Solution :
Let x be the measure of the angle.
The measure of a complement and a supplement of the angle x are (90° - x) and (180° - x) respectively.
Given : The measure of a complement of an angle ever equal exactly half the measure of a supplement of the angle.
Multiply both sides by 2.
2(90° - x) = 180° - x
180° - 2x = 180° - x
Subtract 180° from both sides.
-2x = -x
Add x to both sides.
-x = 0°
Multiply both sides by -1.
x = 0°
Conclusion :
Yes, the measure of a complement of an angle ever equal exactly half the measure of a supplement of the angle and that angle is 0°.
Problem 7 :
The m<A is complementary to the <B. The m<C is complementary to m<B. If m<A = 62°, what is m<B and the m<C.
Solution :
Since <A and <B are complementary angles,
<A + <B = 90°
Given that, m<A = 62°
62° + <B = 90
<B = 90 - 62
<B = 18°
Since m<B and m<C are complementary, then
18° + m<C = 90
m<C = 90 - 18
m<C = 62°
Problem 8 :
The m<D is supplementary to the m<E. The m<F is supplementary to m<E. If m<F = 113°, what is m<D and the m<E.
Solution :
Since <D and <E are supplementary angles,
m<D + m<E = 180° -----(1)
m<F + m<E = 180°-----(2)
Applying m<F = 113° in (2), we get
113° + m<E = 180°
m<E = 180° - 113°
m<E = 67°
By applying m<E = 67° in (1), we get
m<D + 67° = 180°
m<D = 180° - 67°
m<D = 113°
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