Question

which angles are supplementary angles? (with a diagram of lines and angles labeled a, b, c, d, e, f, g, h) options: ∠efc and ∠gfh; ∠bca and ∠bcf; ∠gfh and ∠dcf; ∠gfc and ∠bcf (note: possible typo in the last option as seen in the image)

Explanation:

Step1: Recall Supplementary Angles Definition

Supplementary angles are two angles whose sum is \(180^\circ\) (a straight angle). They can be adjacent (forming a linear pair) or non - adjacent.

Step2: Analyze Each Option

• Option 1: \(\angle EFC\) and \(\angle GFH\)
• \(\angle EFC\) and \(\angle GFH\) are vertical angles (opposite angles formed by two intersecting lines). Vertical angles are equal, not supplementary (unless each is \(90^\circ\), but there's no indication of that here). So this pair is not supplementary.
• Option 2: \(\angle BCA\) and \(\angle BCF\)
• \(\angle BCA\) and \(\angle BCF\) form a linear pair (they are adjacent and their non - common sides form a straight line). By the definition of a linear pair, their sum is \(180^\circ\), so they are supplementary.
• Option 3: \(\angle GFH\) and \(\angle DCF\)
• \(\angle GFH\) and \(\angle DCF\): There is no relationship (like linear pair, same - side interior angles etc.) that would make their sum \(180^\circ\). From the diagram, these angles are not related in a way to be supplementary.
• Option 4: \(\angle GFC\) and \(\angle BCF\)
• \(\angle GFC\) and \(\angle BCF\): These are alternate interior angles (since \(BD\parallel EG\) and \(AH\) is a transversal). Alternate interior angles are equal, not supplementary (unless each is \(90^\circ\), which is not indicated).

Answer:

\(\angle BCA\) and \(\angle BCF\)