Sure, let's find the angle step-by-step.
1. Identify the Complement Relationship:
Recall that the sum of an angle and its complement is always [tex]\( 90^\circ \)[/tex]. If [tex]\( x \)[/tex] represents the complement of the angle, then the angle itself can be expressed as [tex]\( 90^\circ - x \)[/tex].
2. Express the Given Condition:
According to the problem, the angle is [tex]\( 21^\circ \)[/tex] more than twice its complement. We can write this relationship algebraically as:
[tex]\[
\text{Angle} = 2x + 21^\circ
\][/tex]
3. Set Up the Equation:
Since [tex]\( x \)[/tex] is the complement of the angle, the angle can also be written as [tex]\( 90^\circ - x \)[/tex]. Thus, we set up the equation:
[tex]\[
90^\circ - x = 2x + 21^\circ
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
90 - x = 2x + 21
\][/tex]
First, combine like terms by adding [tex]\( x \)[/tex] to both sides:
[tex]\[
90 = 3x + 21
\][/tex]
Then, subtract 21 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
69 = 3x
\][/tex]
Finally, divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 23
\][/tex]
5. Find the Angle:
Now that we have the value of the complement [tex]\( x \)[/tex], which is [tex]\( 23^\circ \)[/tex], we can find the actual angle using the expression [tex]\( 90^\circ - x \)[/tex]:
[tex]\[
\text{Angle} = 90^\circ - 23^\circ = 67^\circ
\][/tex]
So, the angle that is [tex]\( 21^\circ \)[/tex] more than twice its complement is [tex]\( 67^\circ \)[/tex].
