< = Angle
<R and <S are complementary angles, and <U and <V are also complementary angles. If m<R=y-2, m<S=2x+3, m<U=2x-y, and m<V=x-1, find the values of x, y, m<R, m<S, m<U, and m<V
y-2 + 2x+3 = 90
2x + y = 89
2x-y + x-1 = 90
3x - y = 91
add the two equations
5x = 180
x = 36
take it from there,
sub x=36 into one of the equations above to find y, then evaluate each of the angles
To find the values of x, y, m<R, m<S, m<U, and m<V, we'll use the information given.
Since <R and <S are complementary angles, their measures add up to 90 degrees.
So, m<R + m<S = 90 degrees.
Similarly, since <U and <V are complementary angles, their measures add up to 90 degrees.
So, m<U + m<V = 90 degrees.
Let's use this information to solve for x and y step-by-step.
1. Start with the given information:
m<R = y - 2
m<S = 2x + 3
m<U = 2x - y
m<V = x - 1
2. Use the information about complementary angles:
m<R + m<S = 90 degrees
y - 2 + 2x + 3 = 90
Simplify the equation:
2x + y + 1 = 90
2x + y = 89 ---- Equation 1
3. Use the information about complementary angles:
m<U + m<V = 90 degrees
2x - y + x - 1 = 90
Simplify the equation:
3x - y - 1 = 90 ---- Equation 2
4. Now we have a system of two equations with two variables (x and y):
Equation 1: 2x + y = 89
Equation 2: 3x - y = 91
Solve the system of equations by either substitution or elimination method.
Let's use the elimination method in this case:
Add the equations together:
(2x + y) + (3x - y) = 89 + 91
5x = 180
Divide both sides by 5:
x = 36
5. Substitute the value of x back into Equation 1:
2(36) + y = 89
72 + y = 89
Subtract 72 from both sides:
y = 17
6. Substitute the values of x and y into the angle measures:
m<R = y - 2 = 17 - 2 = 15
m<S = 2x + 3 = 2(36) + 3 = 75
m<U = 2x - y = 2(36) - 17 = 55
m<V = x - 1 = 36 - 1 = 35
Therefore, the values are:
x = 36
y = 17
m<R = 15
m<S = 75
m<U = 55
m<V = 35
To find the values of x, y, m<R, m<S, m<U, and m<V, we need to use the given information about the angles <R, <S, <U, and <V.
First, let's look at the definition of complementary angles. Complementary angles are two angles that add up to 90 degrees.
We know that <R and <S are complementary angles, so we can set up the equation:
m<R + m<S = 90 (Equation 1)
Similarly, <U and <V are also complementary angles, so we can set up another equation:
m<U + m<V = 90 (Equation 2)
Next, we can use the given relationships between the angles and variables to substitute the values into these equations.
From the given information, we have:
m<R = y - 2 (Equation 3)
m<S = 2x + 3 (Equation 4)
m<U = 2x - y (Equation 5)
m<V = x - 1 (Equation 6)
Substituting equations 3 and 4 into equation 1, we get:
(y - 2) + (2x + 3) = 90
Simplifying the equation, we have:
y + 2x + 1 = 90
Next, we can substitute equations 5 and 6 into equation 2:
(2x - y) + (x - 1) = 90
Simplifying the equation, we have:
3x - y - 1 = 90
Now, we have a system of two equations with two variables (x and y). We can solve this system to find the values of x and y.
Let's solve the first equation (y + 2x + 1 = 90) for y:
y = 90 - 2x - 1
y = 89 - 2x (Equation 7)
Substitute equation 7 into the second equation (3x - y - 1 = 90):
3x - (89 - 2x) - 1 = 90
Simplify the equation:
3x - 89 + 2x - 1 = 90
5x - 90 = 90
5x = 180
x = 36
Now that we have the value of x, substitute it back into equation 7 to find y:
y = 89 - 2(36)
y = 89 - 72
y = 17
Therefore, the values of x, y, m<R, m<S, m<U, and m<V are:
x = 36,
y = 17,
m<R = y - 2 = 17 - 2 = 15,
m<S = 2x + 3 = 2(36) + 3 = 75,
m<U = 2x - y = 2(36) - 17 = 55,
m<V = x - 1 = 36 - 1 = 35.