Question

practice problems: solve for the given variable and justify.
a.
(diagram with angles 2x + 2° and 5x - 10° formed by a transversal with parallel lines)
b.
(diagram with angles 5x + 10° and 3x + 30° formed by a transversal with parallel lines)

Explanation:

Part a:

Step 1: Identify Angle Relationship

The angles \(2x + 27^\circ\) and \(5x - 10^\circ\) are alternate interior angles (since lines are parallel, alternate interior angles are equal). So, set them equal:
\(2x + 27 = 5x - 10\)

Step 2: Solve for \(x\)

Subtract \(2x\) from both sides:
\(27 = 3x - 10\)

Add \(10\) to both sides:
\(37 = 3x\)

Divide by \(3\):
\(x=\frac{37}{3}\approx12.33^\circ\) (Wait, no—wait, maybe they are same - side? Wait, no, the diagram: if the transversal cuts parallel lines, maybe they are supplementary? Wait, no, let's re - check. Wait, maybe I misidentified. Wait, if the two angles are on a straight line? No, the arrows: maybe they are same - side interior angles? Wait, no, the labels: \(2x + 27\) and \(5x - 10\) are on a transversal with parallel lines. Wait, maybe they are supplementary? Wait, no, let's do again. Wait, maybe the angles are supplementary (same - side interior angles). So \(2x + 27+5x - 10 = 180\).

So \(7x + 17 = 180\)

\(7x=180 - 17=163\)

\(x=\frac{163}{7}\approx23.29\). Wait, I think I made a mistake earlier. Let's look at the diagram again. The two angles are on a transversal, with the parallel lines. If the angles are same - side interior angles, they are supplementary. So:

\(2x + 27+5x - 10 = 180\)

\(7x + 17 = 180\)

\(7x=163\)

\(x=\frac{163}{7}\approx23.29\). Wait, maybe the first assumption was wrong. Alternatively, if they are alternate exterior? No, the diagram shows the angles between the parallel lines. Let's proceed with the correct approach. Let's assume they are supplementary (same - side interior angles). So:

\(2x + 27+5x - 10 = 180\)

\(7x + 17 = 180\)

\(7x = 163\)

\(x=\frac{163}{7}\approx23.29\). But maybe the angles are equal (alternate interior). Let's check: \(2x + 27 = 5x - 10\)

\(3x=37\)

\(x=\frac{37}{3}\approx12.33\). The correct relationship depends on the diagram. Since the arrows are on the lines, the transversal creates angles. If the angles are on alternate sides, equal; same - side, supplementary. Let's assume alternate interior (equal). So \(2x + 27 = 5x - 10\)

\(3x = 37\)

\(x=\frac{37}{3}\approx12.33\).

Part b:

Step 1: Identify Angle Relationship

The angles \(5x + 10^\circ\) and \(3x + 30^\circ\) are alternate interior angles (parallel lines, so equal). So set them equal:
\(5x + 10 = 3x + 30\)

Step 2: Solve for \(x\)

Subtract \(3x\) from both sides:
\(2x + 10 = 30\)

Subtract \(10\) from both sides:
\(2x = 20\)

Divide by \(2\):
\(x = 10\)

Answer:

Part a:

If angles are alternate interior (equal): \(x=\boldsymbol{\frac{37}{3}}\) (or \(\approx12.33^\circ\)); if supplementary: \(x=\boldsymbol{\frac{163}{7}}\) (or \(\approx23.29^\circ\)) (depends on diagram).

Part b:

\(x=\boldsymbol{10}\) (since \(5x + 10 = 3x + 30\) gives \(x = 10\))