the ratio of the measure of the complementsof two angle is3:4while the ratio of the measure of their supplements is 9:10 find the measure of each angle.
let one angle be x
let the other be y
then their complements are 90-x and 90-y
and their supplements: 180-x and 180-y
(90-x)/(90-y) = 3/4
270 - 3y = 360 - 4x
4x - 3y = 90 , #1
(180-x)/(180-y) = 9/10
1620 - 9y = 1800 - 10x
10x - 9y = 180 , #2
#1 times 3
12x - 9y = 270
subtract #2
2x = 90
x = 45
into #1
180 - 3y = 90
-3y = -90
y = 30
so one angle is 45° and the other is 30°
check:
their complements are 45 and 60
45 : 60 = 3:4
their supplements are 135 and 150
135:150 = 9:10
YEAHHH
the ratio of the complement of an angle to the supplement is 1:7.Find the angle.
To solve this problem, let's first define the angles. Let's call the first angle A and the second angle B.
The complement of an angle A is 90 degrees minus the measure of angle A. So, the measure of the complement of angle A is 90 - A degrees, and the measure of the complement of angle B is 90 - B degrees.
Given that the ratio of the measures of the complements of the two angles is 3:4, we can set up the following equation:
(90 - A) : (90 - B) = 3 : 4
Similarly, the supplement of an angle A is 180 degrees minus the measure of angle A. So, the measure of the supplement of angle A is 180 - A degrees, and the measure of the supplement of angle B is 180 - B degrees.
Given that the ratio of the measures of the supplements of the two angles is 9:10, we can set up the following equation:
(180 - A) : (180 - B) = 9 : 10
Now we have two equations:
(90 - A) : (90 - B) = 3 : 4 ...(1)
(180 - A) : (180 - B) = 9 : 10 ...(2)
To find the measure of each angle, we can solve these equations simultaneously. Here's how:
First, we need to express the ratios in terms of the same variable. We can do this by cross-multiplying the ratios:
4 * (90 - A) = 3 * (90 - B) ...(3)
10 * (180 - A) = 9 * (180 - B) ...(4)
Now, let's simplify equations (3) and (4):
360 - 4A = 270 - 3B
900 - 10A = 1620 - 9B
Next, let's simplify these equations further:
-4A + 3B = -90 ...(5)
-10A + 9B = -720 ...(6)
Now, we can solve these two equations using any method of solving simultaneous equations. Let's use the method of substitution:
From equation (5), we can express A in terms of B:
-4A = -90 - 3B
A = (90 + 3B)/4 ...(7)
Substitute equation (7) into equation (6):
-10 * ((90 + 3B)/4) + 9B = -720
Simplify this equation:
-900 - 30B + 36B = -2880
6B = 1980
B = 330
Now substitute the value of B into equation (7):
A = (90 + 3 * 330)/4
A = (90 + 990)/4
A = 1080/4
A = 270
Therefore, the measure of angle A is 270 degrees and the measure of angle B is 330 degrees.