Question

The angle measurements in the diagram are represented by the following expressions.
\[\qquad \blueD{\angle A=5x -12^\circ} \]
\[\qquad \green{\angle B=2x + 24^\circ} \]
Two parallel lines with a third line intersecting each line. Where the third line crosses the left most parallel line, the top, right most angle measure has a curved blue line and is labeled A. Where the third line crosses the right most parallel line, the bottom, left angle measure has a curved green line and is labeled B.
\[A\]
\[B\]
Solve for
\[x\] and then find the measure of
\[\greenD{\angle B}\]:
\[\greenD{\angle B} = \]

\[^\circ\]

Answer 1

Since angles \( A \) and \( B \) are formed by parallel lines and a transversal, they are alternate interior angles. Therefore, we can set their measures equal to each other:

\[
5x - 12 = 2x + 24
\]

Now, let's solve for \( x \).

1. Start by isolating \( x \). Subtract \( 2x \) from both sides:

\[
5x - 2x - 12 = 24
\]

2. This simplifies to:

\[
3x - 12 = 24
\]

3. Next, add \( 12 \) to both sides:

\[
3x = 36
\]

4. Now, divide both sides by \( 3 \):

\[
x = 12
\]

Now that we have the value of \( x \), we can find the measure of angle \( B \).

To find \( \angle B \), substitute \( x = 12 \) back into the expression for \( \angle B \):

\[
\angle B = 2x + 24
\]

Substituting \( x \):

\[
\angle B = 2(12) + 24
\]

Calculating this gives:

\[
\angle B = 24 + 24 = 48^\circ
\]

Thus, the measure of \( \angle B \) is:

\[
\greenD{\angle B} = 48^\circ
\]

So the final answers are:

\[
x = 12
\]

\[
\greenD{\angle B} = 48^\circ
\]