Question

complete the paragraph proof.
we are given that $overrightarrow{eb}$ bisects $angle aec$. from the diagram, $angle ced$ is a right angle, which measures 90 degrees. since the measure of a straight angle is $180^circ$, the measure of angle dropdown must also be $90^circ$ by the dropdown. a bisects the angle measure in half. $mangle aeb = 45^circ$.

Explanation:

Step1: Analyze the straight angle

A straight angle (like ∠AED) is \(180^\circ\). ∠CED is \(90^\circ\), so the remaining angle (∠AEB + ∠BEC or ∠AEC) and ∠CED should add to \(180^\circ\). Wait, no—since ∠CED is \(90^\circ\), the angle adjacent to it on the straight line (∠AEC? No, wait, ∠AED is straight. So ∠AED = ∠AEB + ∠BEC + ∠CED? No, maybe the diagram has E with rays: A---E---D? No, the bisector is EB, and ∠CED is right. So ∠AED is straight (\(180^\circ\)), ∠CED is \(90^\circ\), so ∠AEC must be \(90^\circ\) (since ∠AEC + ∠CED = \(180^\circ\)? Wait, no, if E is the vertex, and rays: A, B, C, D. So EB bisects ∠AEC. ∠CED is \(90^\circ\), and ∠AED is straight (\(180^\circ\)), so ∠AEC = \(180^\circ - 90^\circ = 90^\circ\)? Wait, no, maybe ∠AED is straight, so ∠AEB + ∠BEC + ∠CED = \(180^\circ\), but ∠CED is \(90^\circ\), so ∠AEB + ∠BEC = \(90^\circ\). But EB bisects ∠AEC, so ∠AEB = ∠BEC. Then each is \(45^\circ\). But the question is about which angle is \(90^\circ\) besides ∠CED. Since ∠AED is straight (\(180^\circ\)) and ∠CED is \(90^\circ\), then ∠AEC must be \(90^\circ\) (because ∠AEC + ∠CED = \(180^\circ\)? Wait, no, if ∠AED is straight, then ∠AEC and ∠CED are supplementary? Wait, maybe the diagram is A---E---D, with B and C between A and D? No, the bisector is EB, so EB splits ∠AEC into two equal angles. ∠CED is \(90^\circ\), so ∠AEC must be \(90^\circ\) because ∠AED is \(180^\circ\), so ∠AEC + ∠CED = \(180^\circ\) → ∠AEC = \(90^\circ\). Wait, but the dropdown has AEB, AEC, BEC, BED. Wait, the sentence is "the measure of angle [dropdown] must also be \(90^\circ\) by the...". Since ∠CED is \(90^\circ\), and ∠AED is straight, then ∠AEC (if E is between A and D, and C is another ray) would be supplementary to ∠CED? Wait, maybe the correct angle is ∠AEC? No, wait, the bisector is EB, so EB splits ∠AEC into two equal parts. Then, if ∠CED is \(90^\circ\), and ∠AED is \(180^\circ\), then ∠AEC is \(90^\circ\)? Wait, no, maybe ∠BED? No, ∠CED is \(90^\circ\), so ∠BED would be ∠BEC + ∠CED. But EB bisects ∠AEC, so ∠AEB = ∠BEC. Let's think again: ∠AED is a straight angle (\(180^\circ\)). ∠CED is \(90^\circ\), so ∠AEC = \(180^\circ - 90^\circ = 90^\circ\)? Wait, no, ∠AEC and ∠CED are adjacent angles forming ∠AED, so yes, ∠AEC + ∠CED = \(180^\circ\), so ∠AEC = \(90^\circ\). But the dropdown options: AEB, AEC, BEC, BED. Wait, the sentence is "the measure of angle [x] must also be \(90^\circ\) by the...". Since ∠CED is \(90^\circ\), and ∠AED is straight, then ∠AEC is \(90^\circ\)? But EB bisects ∠AEC, so ∠AEB = ∠BEC = \(45^\circ\). Wait, maybe the correct angle is ∠AEC? No, wait, maybe the angle is ∠BED? No, ∠CED is \(90^\circ\), ∠BEC is \(45^\circ\), so ∠BED is \(135^\circ\). No. Wait, the first part: ∠CED is \(90^\circ\). Then, since ∠AED is \(180^\circ\), the angle ∠AEB + ∠BEC (which is ∠AEC) plus ∠CED is \(180^\circ\). Wait, no, maybe the diagram is E with rays: A, B, C, D, where B is between A and C, and C is between B and D. So ∠AED is straight (A to E to D). ∠CED is \(90^\circ\) (C to E to D is right). Then ∠AEC is A to E to C, which is adjacent to ∠CED, so ∠AEC + ∠CED = \(180^\circ\), so ∠AEC = \(90^\circ\). Then EB bisects ∠AEC, so ∠AEB = ∠BEC = \(45^\circ\). So the angle that is \(90^\circ\) besides ∠CED is ∠AEC? Wait, but the dropdown has AEC as an option. Wait, the sentence is "the measure of angle [dropdown] must also be \(90^\circ\) by the...". So the answer should be AEC? Wait, no, maybe the angle is ∠BED? No, let's check the options again. The dropdown options are AEB, AEC, BEC, BED. The correct answer is AEC? Wait, no, maybe the angle is ∠AEC? Wait, the problem says "since the measure of a straight angle is \(180^\circ\), the measure of angle [x] must also be \(90^\circ\) by the...". So ∠CED is \(90^\circ\), and ∠AED is straight, so ∠AEC (which is adjacent to ∠CED) is \(90^\circ\) because \(180 - 90 = 90\). So the angle is ∠AEC. Wait, but the bisector is EB, so EB splits ∠AEC into two \(45^\circ\) angles. So the correct option is AEC? Wait, no, maybe the angle is ∠BED? No, ∠BED would be ∠BEC + ∠CED = \(45 + 90 = 135\). No. Wait, maybe the angle is ∠AEC? Let's confirm: ∠AED is straight (\(180^\circ\)), ∠CED is \(90^\circ\), so ∠AEC = \(180 - 90 = 90^\circ\). Yes, that makes sense. So the answer is AEC.

Answer:

AEC