Question

find the value of the variable.
x =
(simplify your answer.)
diagram: two intersecting lines (ab and cd) intersecting at point e. angle ∠aed is labeled ( 3x^circ ), angle ∠bec is labeled ( (84 - x)^circ ). points a, b are on one line; c, d are on the other line.

Explanation:

Answer:

To solve for \( x \), we recognize that \( \angle AED \) and \( \angle BEC \) are vertical angles, so they are equal. Wait, actually, looking at the diagram, the angles \( 3x^\circ \) and \( (84 - x)^\circ \) are vertical angles? Wait, no, maybe they are supplementary? Wait, no, if two lines intersect, vertical angles are equal. Wait, let's check: if lines \( AB \) and \( CD \) intersect at \( E \), then \( \angle AED \) and \( \angle BEC \) are vertical angles, so they should be equal. Wait, but in the diagram, the angles are \( 3x \) and \( 84 - x \). Wait, maybe they are vertical angles, so \( 3x = 84 - x \)? Wait, no, that would give \( 4x = 84 \), so \( x = 21 \). Wait, let's solve:

\( 3x = 84 - x \)

Add \( x \) to both sides:

\( 4x = 84 \)

Divide both sides by 4:

\( x = 21 \)

Wait, but maybe they are supplementary? Wait, no, vertical angles are equal. So the correct equation is \( 3x = 84 - x \), so solving gives \( x = 21 \).

Wait, but let's confirm. If \( x = 21 \), then \( 3x = 63 \) and \( 84 - x = 63 \), so they are equal, which makes sense for vertical angles. So the value of \( x \) is 21.

\( x = 21 \)