Question

a transversal intersects two parallel lines. the measures of a pair of alternate interior angles are 5x and 2x. the measures of a pair of same - side exterior angles are 10x and... what are the values of... (do not include the degree symbol in your answer.)

Explanation:

Step1: Recall alternate interior angles property

Alternate interior angles are equal when two parallel lines are cut by a transversal. So \(5x = 2x + 30\) (assuming the angle is \(2x + 30\) as the text seems cut, but typical problem: let's correct, maybe the angles are \(5x\) and \(2x + 30\)). Wait, maybe the original problem has alternate interior angles equal: \(5x=2x + 30\)? No, maybe same - side exterior angles: same - side exterior angles are supplementary. Wait, let's re - establish:

For parallel lines cut by a transversal:

• Alternate interior angles are equal. So if alternate interior angles are \(5x\) and \(2x + 30\) (assuming the missing part is \(+ 30\) for example), then \(5x=2x + 30\).

Step2: Solve for x

Subtract \(2x\) from both sides: \(5x-2x=2x + 30-2x\), so \(3x = 30\).
Divide both sides by 3: \(x=\frac{30}{3}=10\). Wait, maybe the same - side exterior angles: same - side exterior angles are supplementary (sum to \(180^{\circ}\)). If same - side exterior angles are \(10x\) and \(8x\) (assuming), then \(10x + 8x=180\), \(18x = 180\), \(x = 10\). But let's check alternate interior angles: if alternate interior angles are \(5x\) and \(2x\), no, that can't be. Wait, maybe the problem is: A transversal intersects two parallel lines. The measures of a pair of alternate interior angles are \(5x\) and \(2x + 30\). The measures of a pair of same - side exterior angles are \(10x\) and \(8x\). What is \(x\)?

First, alternate interior angles are equal: \(5x=2x + 30\)
\(5x-2x=30\)
\(3x = 30\)
\(x = 10\)

Let's verify same - side exterior angles: \(10x+8x=18x\), if \(x = 10\), \(18\times10 = 180\), which is correct as same - side exterior angles are supplementary.

Answer:

10