find the angle which is four times its complement
Find the angle which is four times its complement
Concept of Complementary Angles
Complementary angles are a pair of angles whose measures add up to 90^\circ. If two angles are complementary, it implies that the sum of their measures is always three right angles.
For example:
- If one angle is 30^\circ, its complement is 60^\circ because 30^\circ + 60^\circ = 90^\circ.
Now, we need to find an angle that is four times its complementary angle.
Step-by-Step Solution
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Define the Angles:
Let’s denote the angle we need to find as x^\circ.
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Determine the Complementary Angle:
Since x^\circ and its complement add up to 90^\circ, the complementary angle would be (90^\circ - x^\circ).
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Set Up the Equation:
According to the problem, the angle x^\circ is four times its complement. Therefore, we can set up the following equation:
$$ x = 4(90 - x) $$ -
Simplify the Equation:
Let’s solve for x step-by-step.
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First, distribute the (4) on the right side:
$$ x = 360 - 4x $$ -
Next, add (4x) to both sides to bring all terms involving (x) to one side:
$$ x + 4x = 360 $$ -
Combine like terms:
$$ 5x = 360 $$
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Solve for (x):
Divide both sides by (5):
$$ x = \frac{360}{5} $$
$$ x = 72 $$Therefore, the angle we need to find is 72^\circ.
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Verify the Solution:
To ensure our solution is correct, let’s verify it:
- The complement of (72^\circ) is (90^\circ - 72^\circ = 18^\circ).
- Check if (72^\circ) is indeed four times its complement:
- (4 \times 18^\circ = 72^\circ).
The conditions of the problem are satisfied, confirming our solution is accurate.
Summary
In conclusion, the angle that is four times its complement is 72^\circ, and its complement is 18^\circ. Solving such problems involves using basic algebraic methods, applying the concept of complementary angles, and ensuring the solution satisfies the given condition. @anonymous6