Question

look at this diagram:
if \\( \overleftrightarrow { d f } \\) and \\( \overleftrightarrow { g i } \\) are parallel lines and \\( m \angle i h j = 126 ^ { \circ } \\), what is \\( m \angle d e c \\)?

Explanation:

Step1: Find the measure of $\angle EHI$

Since $\angle IHJ$ and $\angle EHI$ are supplementary angles (they form a linear - pair), and the sum of supplementary angles is $180^{\circ}$.
If $m\angle IHJ = 126^{\circ}$, then $m\angle EHI=180^{\circ}-m\angle IHJ$.
$m\angle EHI = 180 - 126=54^{\circ}$.

Step2: Use the property of parallel lines

Because $DF\parallel GI$ and $CJ$ is a transversal, $\angle DEC$ and $\angle EHI$ are alternate - interior angles.
Alternate - interior angles formed by two parallel lines and a transversal are congruent. So $m\angle DEC=m\angle EHI$.

Answer:

$54$