Question

in the following diagram, \\( \overline { h i } \\) is parallel to \\( \overline { j k } \\).
what is the measure of \\( \angle x \\)?
angles are not necessarily drawn to scale.

\\( \angle x = \\) \\( \circ \\)

Explanation:

Step1: Find the exterior angle of triangle \( \triangle AHI \)

According to the exterior - angle theorem of a triangle, the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
For \( \triangle AHI \), if we consider the exterior angle at \( I \) (let's call it \( \angle HIK\)), then \( \angle HIK=\angle A+\angle AHI \).
Given \( \angle A = 66^{\circ}\) and \( \angle AHI=56^{\circ}\).
So, \( \angle HIK=66^{\circ}+56^{\circ}\).

Step2: Calculate the value of \( x \)

Since \( \overline{HI}\parallel\overline{JK}\), \( \angle x=\angle HIK\) (corresponding angles, when two parallel lines are cut by a transversal, corresponding angles are equal).
\( \angle x=66 + 56\).

Answer:

\(122\)