∠*K* and ∠*L* are complementary angles.  If *m*∠*K* = (3x+3) and *m*∠*L* = (10x – 4), find the measure of ∠*K*  and ∠*L*

Complementary angles are two angles whose measures add up to \( 90^\circ \). Therefore, we can set up the equation:

\[
m \angle K + m \angle L = 90^\circ
\]

Substitute the given expressions for \( m \angle K \) and \( m \angle L \):

\[
(3x + 3) + (10x - 4) = 90
\]

Now, combine like terms:

\[
3x + 10x + 3 - 4 = 90
\]

\[
13x - 1 = 90
\]

Next, add 1 to both sides:

\[
13x = 91
\]

Now, divide both sides by 13:

\[
x = 7
\]

Now that we have the value of \( x \), we can find the measures of \( m \angle K \) and \( m \angle L \):

\[
m \angle K = 3x + 3 = 3(7) + 3 = 21 + 3 = 24^\circ
\]

\[
m \angle L = 10x - 4 = 10(7) - 4 = 70 - 4 = 66^\circ
\]

So, the measure of \( m \angle K \) is \( 24^\circ \) and the measure of \( m \angle L \) is \( 66^\circ \).

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. If \( m \angle A = 2x + 5 \) and \( m \angle B = 4x - 7 \) are complementary angles, find \( m \angle A \) and \( m \angle B \).
2. Explain the difference between complementary and supplementary angles.
3. If two angles are complementary and one of them measures \( 40^\circ \), what is the measure of the other angle?
4. How can you determine if two angles are complementary if you only know their algebraic expressions?
5. Find the value of \( x \) if \( m \angle P = 8x + 2 \) and \( m \angle Q = 5x - 3 \) are complementary angles.

**Tip:** Always remember to check if angles are complementary (sum to \( 90^\circ \)) or supplementary (sum to \( 180^\circ \)) when solving for unknowns!

Learn how to find the measures of complementary angles using algebraic equations. Solve the problem where angles ∠K and ∠L are complementary, with given expressions for their measures. Understand step-by-step how to determine the measures of angles in relation to their sum of 90 degrees. Suitable for Grades 7-9.

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