Question

which sets of angles listed are supplementary in this diagram?
∠def and ∠feg, ∠bef and ∠aef
∠bed and ∠dea, ∠aec and ∠bec
∠ceg and ∠bec, ∠aed and ∠aeg
∠bed and ∠def, ∠aec and ∠aef

Explanation:

Step1: Recall Supplementary Angles Definition

Supplementary angles are two angles whose sum is \(180^\circ\) (a straight angle). We need to check each option's angle pairs.

Step2: Analyze Option 1 (\(\angle DEF\) & \(\angle FEG\), \(\angle BEF\) & \(\angle AEF\))

- \(\angle DEF + \angle FEG\): Visually, they don't form a straight line. Likely sum not \(180^\circ\).
- \(\angle BEF + \angle AEF\): These two angles form a straight line (since \(A - E - B\) is a straight line), so their sum is \(180^\circ\). But the first pair fails, so option 1 is out.

Step3: Analyze Option 2 (\(\angle BED\) & \(\angle DEA\), \(\angle AEC\) & \(\angle BEC\))

- \(\angle BED + \angle DEA\): \(B - E - A\) is a straight line, so their sum is \(180^\circ\) (supplementary).
- \(\angle AEC + \angle BEC\): \(A - E - B\) is a straight line, and \(\angle AEC + \angle BEC\) (since \(C - E - B\) has a right angle, but wait, \(A - E - B\) is straight, and \(\angle AEC\) and \(\angle BEC\): Wait, \(AEC\) and \(BEC\): Wait, \(A - E - B\) is straight, \(\angle AEC\) and \(\angle BEC\): Wait, actually, \(A - E - B\) is a straight line, and \(\angle AEC + \angle BEC\): Wait, no, \(C - E - B\) is a right angle? Wait, no, the diagram has a right angle at \(E\) between \(C - E - B\)? Wait, no, the right angle is between \(C - E - B\)? Wait, the diagram shows a right angle (square) at \(E\) between \(C\) and \(B\), so \(\angle BEC = 90^\circ\), and \(\angle AEC\): since \(A - E - B\) is straight (\(180^\circ\)), \(\angle AEC + \angle BEC = \angle AEB = 180^\circ\)? Wait, no, \(\angle AEB\) is straight (\(180^\circ\)), and \(\angle BEC\) is \(90^\circ\), so \(\angle AEC = 90^\circ\), so \(\angle AEC + \angle BEC = 90^\circ + 90^\circ = 180^\circ\)? Wait, no, \(\angle AEC\) and \(\angle BEC\): if \(C\) is above \(E\), \(B\) is to the right, \(A\) is to the left, then \(\angle AEC\) is left and up, \(\angle BEC\) is right and up, and together they make \(\angle AEB\) which is straight? Wait, no, \(A - E - B\) is horizontal straight line. \(\angle AEC\) is angle between \(A - E - C\), \(\angle BEC\) is between \(B - E - C\). Since \(\angle BEC\) is \(90^\circ\) (right angle), \(\angle AEC\) is also \(90^\circ\) (since \(A - E - B\) is straight, \(180^\circ\), so \(180^\circ - 90^\circ = 90^\circ\) for \(\angle AEC\)). So \(\angle AEC + \angle BEC = 90^\circ + 90^\circ = 180^\circ\) (supplementary). And \(\angle BED + \angle DEA\): \(B - E - D\) and \(D - E - A\): \(B - E - A\) is straight, so \(\angle BED + \angle DEA = \angle BEA = 180^\circ\) (supplementary). Wait, but let's check other options.

Wait, maybe I made a mistake. Let's re-examine.

Wait, option 2: \(\angle BED\) and \(\angle DEA\): \(B - E - D\) and \(D - E - A\). Since \(B - E - A\) is a straight line, \(\angle BED + \angle DEA = \angle BEA = 180^\circ\) (supplementary). Then \(\angle AEC\) and \(\angle BEC\): \(A - E - C\) and \(B - E - C\). Since \(A - E - B\) is straight (\(180^\circ\)), and \(\angle BEC = 90^\circ\) (right angle), \(\angle AEC = 90^\circ\), so \(90^\circ + 90^\circ = 180^\circ\) (supplementary). So both pairs in option 2 are supplementary? Wait, no, wait \(\angle BED\) and \(\angle DEA\): \(B - E - D\) is a line? No, \(D\) is below \(E\), \(B\) is to the right, \(A\) is to the left. So \(\angle BED\) is between \(B - E - D\) (down and right), \(\angle DEA\) is between \(D - E - A\) (down and left). Together, they form \(B - E - A\) (straight line), so their sum is \(180^\circ\) (supplementary). And \(\angle AEC\) (left and up) and \(\angle BEC\) (right and up) form \(A - E - B\) (straight line), sum \(180^\circ\). So option 2's pairs are both supplementary. Wait, but let's check other options.

Step4: Analyze Option 3 (\(\angle CEG\) & \(\angle BEC\), \(\angle AED\) & \(\angle AEG\))

- \(\angle CEG + \angle BEC\): \(\angle BEC\) is \(90^\circ\), \(\angle CEG\) is some angle, sum not \(180^\circ\) (since \(CEG\) is part of \(CEB\) maybe? No, \(G\) is another ray. So likely not supplementary.
- \(\angle AED + \angle AEG\): Unlikely to sum to \(180^\circ\). So option 3 is out.

Step5: Analyze Option 4 (\(\angle BED\) & \(\angle DEF\), \(\angle AEC\) & \(\angle AEF\))

- \(\angle BED + \angle DEF\): These two angles are adjacent but not forming a straight line. Sum not \(180^\circ\).
- \(\angle AEC + \angle AEF\): Also not forming a straight line. So option 4 is out.

Wait, but earlier analysis of option 2: \(\angle BED\) and \(\angle DEA\) (sum to \(180^\circ\) as they form \(A - E - B\) straight line), and \(\angle AEC\) and \(\angle BEC\) (sum to \(180^\circ\) as \(A - E - B\) is straight and \(\angle AEC + \angle BEC = \angle AEB = 180^\circ\) since \(\angle BEC = 90^\circ\) and \(\angle AEC = 90^\circ\)). So option 2's pairs are both supplementary.

Answer:

\(\boldsymbol{\angle BED}\) and \(\boldsymbol{\angle DEA}\), \(\boldsymbol{\angle AEC}\) and \(\boldsymbol{\angle BEC}\) (the second option in the list)