Question

lines k, l, m with angles (3x + 10)° and (5x + 18)° as shown in the image.

Explanation:

Assuming lines \( l \) and \( m \) are parallel, and the angles \( (3x + 10)^\circ \) and \( (5x + 18)^\circ \) are same - side interior angles (or alternate interior angles? Wait, no, looking at the diagram, if \( l\parallel m \), then the two angles should be supplementary if they are same - side interior angles, or equal if they are alternate interior angles? Wait, no, let's re - examine. Wait, the angle \( (3x + 10)^\circ \) and \( (5x + 18)^\circ \): if \( l\parallel m \), then these two angles are same - side interior angles, so they should be supplementary? Wait, no, maybe alternate interior angles? Wait, no, the transversal is the line \( k \). Wait, actually, if \( l\parallel m \), then the angle \( (3x + 10)^\circ \) and \( (5x + 18)^\circ \) are same - side interior angles, so their sum is \( 180^\circ \)? Wait, no, maybe I made a mistake. Wait, let's assume that \( l\parallel m \), and the two angles are same - side interior angles, so:

Step 1: Set up the equation

Since \( l\parallel m \) and the two angles are same - side interior angles, they are supplementary. So we have the equation:
\( (3x + 10)+(5x + 18)=180 \)

Step 2: Combine like terms

Combine the \( x \) terms and the constant terms:
\( 3x+5x + 10 + 18=180 \)
\( 8x+28 = 180 \)

Step 3: Solve for \( x \)

Subtract 28 from both sides:
\( 8x=180 - 28 \)
\( 8x=152 \)
Divide both sides by 8:
\( x=\frac{152}{8}=19 \)

Wait, but maybe the angles are alternate interior angles? If they are alternate interior angles, then \( 3x + 10=5x+18 \), but \( 3x + 10=5x + 18\) gives \( - 2x=8\), \( x=- 4\), which is not possible for an angle measure. So they must be same - side interior angles.

Now, let's find the measure of the angles.

For \( 3x + 10 \): substitute \( x = 19 \), we get \( 3\times19+10=57 + 10 = 67^\circ \)

For \( 5x + 18 \): substitute \( x = 19 \), we get \( 5\times19+18=95 + 18 = 113^\circ \)

And \( 67+113 = 180 \), which checks out.

But maybe the problem is to find \( x \). Let's assume the problem is to find \( x \) when \( l\parallel m \).

Answer:

If we are to find \( x \), then \( x = 19 \)