To solve the given problem, let's go through it step-by-step.
1. Understanding Complementary Angles:
Complementary angles are two angles whose measures add up to 90 degrees. This means:
[tex]\[
\text{angle J} + \text{angle K} = 90^\circ
\][/tex]
2. Assign the Given Values:
We are given:
[tex]\[
\text{angle J} = (3x + 15)^\circ
\][/tex]
[tex]\[
\text{angle K} = 63^\circ
\][/tex]
3. Set up the Equation:
Since angles J and K are complementary:
[tex]\[
(3x + 15) + 63 = 90
\][/tex]
4. Simplify the Equation:
Combine like terms to simplify:
[tex]\[
3x + 15 + 63 = 90
\][/tex]
[tex]\[
3x + 78 = 90
\][/tex]
5. Isolate the Variable:
Subtract 78 from both sides to isolate the term with x:
[tex]\[
3x = 90 - 78
\][/tex]
[tex]\[
3x = 12
\][/tex]
6. Solve for x:
Divide both sides by 3 to solve for x:
[tex]\[
x = \frac{12}{3}
\][/tex]
[tex]\[
x = 4
\][/tex]
7. Verifying the Solution:
Now, let’s verify by finding the measure of angle J.
[tex]\[
\text{angle J} = 3x + 15 = 3(4) + 15 = 12 + 15 = 27^\circ
\][/tex]
8. Conclusion:
Therefore, the value of x is [tex]\( 4 \)[/tex]. When x is 4, the measure of angle J is [tex]\( 27^\circ \)[/tex], and angle K is [tex]\( 63^\circ \)[/tex]. Both angles are complementary as their sum is:
[tex]\[
27^\circ + 63^\circ = 90^\circ
\][/tex]
Thus, the solution confirms that our original calculations are correct.
