To find the measure of each angle given that [tex]\( \angle K \)[/tex] and [tex]\( \angle L \)[/tex] are complementary angles, we need to follow these steps:
1. Understand the Concept of Complementary Angles:
Complementary angles are two angles whose measures add up to 90 degrees. Therefore, we can write the equation:
[tex]\[
m \angle K + m \angle L = 90^\circ
\][/tex]
2. Substitute the Expressions for the Angles:
It is given that:
[tex]\[
m \angle K = (3x + 3)^\circ
\][/tex]
[tex]\[
m \angle L = (10x - 4)^\circ
\][/tex]
Substituting these into the complementary angle equation, we get:
[tex]\[
(3x + 3) + (10x - 4) = 90
\][/tex]
3. Combine Like Terms and Solve for [tex]\( x \)[/tex]:
Simplify the equation by combining like terms:
[tex]\[
3x + 10x + 3 - 4 = 90
\][/tex]
[tex]\[
13x - 1 = 90
\][/tex]
Add 1 to both sides of the equation:
[tex]\[
13x = 91
\][/tex]
Divide both sides by 13:
[tex]\[
x = 7
\][/tex]
4. Calculate the Measure of Each Angle:
With [tex]\( x = 7 \)[/tex], we can substitute this value back into the expressions for [tex]\( \angle K \)[/tex] and [tex]\( \angle L \)[/tex]:
For [tex]\( \angle K \)[/tex]:
[tex]\[
m \angle K = 3x + 3 = 3(7) + 3 = 21 + 3 = 24^\circ
\][/tex]
For [tex]\( \angle L \)[/tex]:
[tex]\[
m \angle L = 10x - 4 = 10(7) - 4 = 70 - 4 = 66^\circ
\][/tex]
5. Verification:
To verify, sum the calculated measures of [tex]\( \angle K \)[/tex] and [tex]\( \angle L \)[/tex] to ensure they add up to 90 degrees:
[tex]\[
24^\circ + 66^\circ = 90^\circ
\][/tex]
Therefore, the measures of the angles are:
[tex]\[
m \angle K = 24^\circ
\][/tex]
[tex]\[
m \angle L = 66^\circ
\][/tex]
