Geometry problem?

Question

1. the measure of a supplement of an angle is 6 times the measure of the complement of the angle. Find the measure of the angle, its suppplement, and its complement.

2. 12 degrees less than one half the supplement of an angle is the same as the complement. find the measure of the angle.

i dont get these...plz solve it and tell me how you did it!

Answer 1

If the measure of an angle is a, then the measure of its complement is 90-a and the measure of its supplement is 180-a, These are the definitions of complement and supplement when you are measuring angles in degrees.

1. Let a be the measure of your angle. The measure of the supplement of the angle is 6 times the measure of its complement, that is, 180 - a = 6 ( 90 - a ) , or 180 - a = 540 - 6a. Add 6a to both sides of this equation. Then 180 - a + 6a = 540 - 6a + 6a, or in other words 180 + 5a = 540 + ( 6a - 6a ) = 540 + 0 = 540. now subtract 180 from both sides. Then 180 - 180 + 5a =540 - 180 or 0 + 5a = 5a = 360. Divide both sides by 5 and you get a = the measure of the angle = 72. Then just use the definitions above to get the measure of the supplement = 180 - 72 = 108, and the measure of the complement = 90 - 72 =18.

To check this notice that the measure of the supplement = 108 = 6 X 18 = 6 times the measure of the complement.

2. Let the measure of the angle = a. One half the supplement of the angle is 1/2 (180 - a), so twelve degrees less than one half the supplement is 1/2 (180 - a) - 12. This is the same as the complement, which is 90 - a, so we have
1/2 (180 - a) - 12 = 90 - a. Now add 12 to both sides. You get
1/2 (180 - a ) - 12 + 12 = 90 + 12 - a, or
1/2 (180 - a ) + 0 = 1/2 (180 - a) = 102 - a. Now multiply both sides by 2: ( 2 )(1/2) (180 - a) = 1 (180 - a ) = 180 - a =
2 (102 - a ) = 204 - 2a. Add 2a to both sides to get 180 - a + 2a = 204 - 2a + 2a, or 180 + a = 204 + 0 = 204. Subtract 180 from both sides. Then 180 - 180 + a = 204 - 180. This means that
0 + a = a = the measure of the angle = 204 - 180 = 24.

Again, to check, notice that the complement = 90 - 24 = 66.
Now the supplement is 180 - 24 = 156, so one half the supplement is ( 1/2) ( 156 ) = 78, and twelve degrees less than one half the supplement is 78 - 12 = 66, which we already know is the complement.

Good luck!

Answer 2

I won't solve it for you, but I will give you a few hints.

A. If an angle complements another angle, that means that the sum of the two angles equals 90 degrees.

B. If an angle supplements another angle, that means that the sum of the two angles equals 180 degrees.

C. Set this up in an equation, where x equals the measure of the supplement!

Answer 3

180 - x = 6(90 - x)
180 - x = 540 - 6x
5x = 360
x = 72

180 - 72 = 108
90 - 72 = 18

Supplement : 108°
Complement : 18°

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2.)
(1/2)(180 - x) - 12 = 90 - x
(1/2)(180 - x) = 102 - x
180 - x = 204 - 2x
x = 24°