Question

which angles are supplementary angles?
∠dca and ∠gfc ∠gfh and ∠bca
∠efc and ∠efh ∠bcf and ∠dca

Explanation:

Step1: Recall Supplementary Angles Definition

Supplementary angles are two angles whose sum is $180^\circ$ (a straight angle). So we need to check which pair of angles forms a straight line (i.e., are adjacent and their non - common sides form a straight line) or have a sum of $180^\circ$.

Step2: Analyze Each Option

• For $\angle DCA$ and $\angle GFC$: These angles are not adjacent and do not form a straight line. From the diagram, their positions do not suggest that their sum is $180^\circ$.
• For $\angle GFH$ and $\angle BCA$: These angles are not related in a way that their sum would be $180^\circ$. They are not adjacent and their positions do not form a straight line.
• For $\angle EFC$ and $\angle EFH$: The two angles share a common side $EF$ and their non - common sides $FC$ and $FH$ form a straight line (since $CH$ is a straight line). So, by the definition of supplementary angles (adjacent angles forming a straight line), $\angle EFC+\angle EFH = 180^\circ$, so they are supplementary.
• For $\angle BCF$ and $\angle DCA$: These angles are vertical angles or related in a way that their sum is not $180^\circ$. They are not adjacent and do not form a straight line.

Answer:

$\angle EFC$ and $\angle EFH$