angle R & angle S are complementary angles..and angle U & and angle V are also complementary angles. if angle R = y-2, angle S = 2x + 3, angle U = 2x-y, and angle V = x-1, find the values of x, y, angle R, S, U, & V
Given that angle R and angle S are complementary angles, we have the equation:
angle R + angle S = 90°
Substituting the given values, we get:
(y-2) + (2x + 3) = 90
Simplifying the equation:
y - 2 + 2x + 3 = 90
Combine like terms:
2x + y + 1 = 90
Now, let's consider angles U and V. Since they are complementary angles, we have:
angle U + angle V = 90°
Substituting the given values, we get:
(2x - y) + (x - 1) = 90
Simplifying the equation:
2x - y + x - 1 = 90
Combine like terms:
3x - y - 1 = 90
To solve for x and y, we have a system of two equations:
2x + y + 1 = 90 ------------ Equation 1
3x - y - 1 = 90 ------------ Equation 2
To solve this system, we can add the two equations together:
(2x + y + 1) + (3x - y - 1) = 90 + 90
Simplifying the equation:
2x + 3x + y - y + 1 - 1 = 180
Combine like terms:
5x = 180
Now solve for x:
x = 180 / 5
x = 36
Substituting this value back into equation 1, we can solve for y:
2(36) + y + 1 = 90
72 + y + 1 = 90
73 + y = 90
Subtracting 73 from both sides:
y = 90 - 73
y = 17
Therefore, the values are:
x = 36
y = 17
Substituting these values back into the given equations, we find:
angle R = y - 2 = 17 - 2 = 15°
angle S = 2x + 3 = 2(36) + 3 = 75°
angle U = 2x - y = 2(36) - 17 = 55°
angle V = x - 1 = 36 - 1 = 35°
So, the values are:
x = 36
y = 17
angle R = 15°
angle S = 75°
angle U = 55°
angle V = 35°
To find the values of x, y, angle R, angle S, angle U, and angle V, we need to use the given information that angle R and angle S are complementary angles, and angle U and angle V are also complementary angles.
Complementary angles are two angles that add up to 90 degrees. So we can set up the following equations:
angle R + angle S = 90 (1)
angle U + angle V = 90 (2)
We are also given the following expressions for each angle:
angle R = y - 2
angle S = 2x + 3
angle U = 2x - y
angle V = x - 1
Let's substitute these expressions into equations (1) and (2):
(y - 2) + (2x + 3) = 90 (3)
(2x - y) + (x - 1) = 90 (4)
Now, let's solve equations (3) and (4) to find the values of x and y:
Equation (3):
y - 2 + 2x + 3 = 90
y + 2x + 1 = 90
y + 2x = 89 (5)
Equation (4):
2x - y + x - 1 = 90
3x - y - 1 = 90
3x - y = 91 (6)
To solve equations (5) and (6), we can use the method of substitution or elimination. Let's solve them by substitution.
From equation (5), we can isolate y:
y = 89 - 2x
Substitute this value of y in equation (6):
3x - (89 - 2x) = 91
3x - 89 + 2x = 91
5x - 89 = 91
5x = 91 + 89
5x = 180
x = 180/5
x = 36
Now substitute the value of x back into equation (5):
y + 2(36) = 89
y + 72 = 89
y = 89 - 72
y = 17
So, the values of x and y are x = 36 and y = 17.
Now, substitute these values of x and y into the expressions for each angle to find the values of the angles:
angle R = y - 2 = 17 - 2 = 15
angle S = 2x + 3 = 2(36) + 3 = 75
angle U = 2x - y = 2(36) - 17 = 55
angle V = x - 1 = 36 - 1 = 35
Therefore, the values of angle R, angle S, angle U, and angle V are:
angle R = 15
angle S = 75
angle U = 55
angle V = 35