Question

lines a and b are parallel and lines e and f are parallel. what is the value of x?

8
82
98
172

Explanation:

Identify the given geometric relationships

We are given that line \(a\) is parallel to line \(b\) (\(a \parallel b\)) and line \(e\) is parallel to line \(f\) (\(e \parallel f\)). We need to find the value of \(x\).

Analyze the angles along line \(a\)

At the intersection of line \(a\) and line \(e\), we see two adjacent angles on a straight line: \(82^\circ\) and \(98^\circ\). These angles are supplementary since \(82^\circ + 98^\circ = 180^\circ\).

Relate angles using parallel lines

Using the Alternate Interior Angles theorem, we can relate the angles formed by the parallel lines \(a\) and \(b\) cut by the transversal line \(e\). The angle of \(98^\circ\) is located in the interior, on the bottom-left of the intersection of line \(a\) and line \(e\).

Solve for \(x\)

The angle \(x^\circ\) is located at the interior, on the bottom-right of the intersection of line \(b\) and line \(e\). Since lines \(a\) and \(b\) are parallel, the consecutive interior angles on the same side of transversal \(e\) must sum to \(180^\circ\), or we can use alternate interior angles to find that the interior angle on the bottom-right of the intersection of line \(b\) and line \(e\) is alternate interior to the top-left angle of \(82^\circ\) at the intersection of line \(a\) and line \(e\). Therefore, we have:

$$x = 82$$

Answer:

• 8
• 82 (Correct answer)
• 98
• 172