Question: The image shows a portion of a staircase railing. There are three parallel rails supported by two posts. The transversal shown is parallel to the posts. What is true about the angles formed by the intersection of railings and the post? Choose one option from each drop-down menu to answer the question. ∠BCD has the same measure as ∠ Choose.... If ∠GFI measures x°, then the measure of ∠HIK is Choose...°. The sum of the measures of ∠CFE and ∠ Choose... is 180°. Graph/shape description: top-left image shows a staircase railing with three parallel slanted rails and two vertical posts; a vertical transversal intersects the rails at points B, C, F, and K on the left post and at D, G, I on the right post, with angle x marked near point F; position_hint ∈ {top-left}.

1. **State the problem:** We have three parallel rails intersected by two vertical posts (transversals). We want to find relationships between the angles formed at these intersections. 2. **Recall angle rules with parallel lines and transversals:** - Corresponding angles are equal. - Alternate interior angles are equal. - Consecutive interior angles (same side interior) sum to 180°. 3. **Analyze ∠BCD and its corresponding angle:** - Since the rails are parallel and the posts are transversals, ∠BCD corresponds to ∠CFG (or ∠GFI depending on labeling). - By the Corresponding Angles Postulate, ∠BCD = ∠CFG. 4. **Analyze ∠GFI and ∠HIK:** - ∠GFI and ∠HIK are corresponding angles formed by the transversal and parallel rails. - Therefore, if ∠GFI = $x^\circ$, then ∠HIK = $x^\circ$. 5. **Sum of ∠CFE and another angle equals 180°:** - ∠CFE and ∠EFG (or ∠EFK depending on labeling) are consecutive interior angles. - Consecutive interior angles sum to 180°. - So, the sum of ∠CFE and ∠EFG is 180°. **Final answers:** - ∠BCD has the same measure as ∠CFG. - If ∠GFI measures $x^\circ$, then the measure of ∠HIK is $x^\circ$. - The sum of the measures of ∠CFE and ∠EFG is 180°.