Find the angle whose supplement is four times its complement

General
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find the angle whose supplement is four times its complement

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Find the angle whose supplement is four times its complement

To solve this problem, we need to understand the definitions of complementary and supplementary angles. Complementary angles are two angles that add up to 90 degrees, while supplementary angles add up to 180 degrees.

Let’s break down the problem:

1. Definitions and Setup

  • Let the angle be denoted as x.
  • The complement of the angle x is 90^\circ - x.
  • The supplement of the angle x is 180^\circ - x.

According to the problem, the supplement of the angle is four times its complement. Thus, we can set up the equation:

180^\circ - x = 4(90^\circ - x)

2. Solving the Equation

Let’s solve the equation step by step:

  1. Expand the Right Side:

    Multiply the complement by 4:

    180^\circ - x = 360^\circ - 4x

  2. Rearrange the Terms:

    Add 4x to both sides of the equation to bring all the x terms to one side:

    180^\circ + 3x = 360^\circ

  3. Isolate x:

    Subtract 180^\circ from both sides to isolate the 3x term:

    3x = 180^\circ

  4. Solve for x:

    Divide both sides by 3 to solve for x:

    x = 60^\circ

3. Verification

It’s always important to verify our solution to ensure it satisfies the original 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.

Now, check if the supplement is indeed four times the complement:

4 \times 30^\circ = 120^\circ

The equation holds true, confirming our solution.

4. Summary

The angle whose supplement is four times its complement is 60^\circ. This solution is derived by setting up the relationship between the angle, its complement, and its supplement, leading to the equation we solved step by step.

@anonymous6