find the angle whose supplement is 4 times its complement
Find the angle whose supplement is 4 times its complement
To solve this problem, we need to understand the relationship between an angle, its supplement, and its complement.
1. Definitions
- Complementary Angles: Two angles are complementary if the sum of their measures is 90^\circ. If one angle is x^\circ, then its complement is 90^\circ - x.
- Supplementary Angles: Two angles are supplementary if the sum of their measures is 180^\circ. If one angle is x^\circ, then its supplement is 180^\circ - x.
Given that the supplement of an angle is 4 times its complement, let’s denote the unknown angle as x^\circ.
2. Setting Up the Equation
The problem states:
[
\text{Supplement} = 4 \times \text{Complement}
]
Using the definitions:
- The supplement of the angle is 180^\circ - x.
- The complement of the angle is 90^\circ - x.
Substitute these into the equation:
[
180^\circ - x = 4 \times (90^\circ - x)
]
3. Solving the Equation
Now, solve for x:
-
Expand the Right Side: Multiply the terms on the right:
[
180^\circ - x = 360^\circ - 4x
] -
Rearrange the Equation: Move the variable terms to one side and constant terms to the other:
[
180^\circ - x + 4x = 360^\circ
] -
Combine Like Terms:
[
3x = 360^\circ - 180^\circ
][
3x = 180^\circ
] -
Solve for x:
[
x = \frac{180^\circ}{3}
][
x = 60^\circ
]
4. Verification
Now, let’s verify that the angle 60^\circ satisfies the given condition:
- The complement of 60^\circ is 90^\circ - 60^\circ = 30^\circ.
- The supplement of 60^\circ is 180^\circ - 60^\circ = 120^\circ.
Check if the supplement is four times the complement:
[
120^\circ = 4 \times 30^\circ
]
Since this equation holds true, the angle x = 60^\circ is indeed correct.
The angle whose supplement is four times its complement is 60^\circ.
By understanding how to set up and solve the equation given the conditions, we’ve found the angle effectively. Keep practicing these steps and equations to become more familiar with such problems. Great work!