Question

given that (overrightarrow{eb}) bisects (angle cea), which statements must be true? select three options. (mangle cea = 90^{circ}) (mangle cef=mangle cea + mangle bef) (mangle ceb = 2(mangle cea)) (angle cef) is a straight angle. (angle aef) is a right angle.

Explanation:

Step1: Analyze the straight - angle property

Since \(C\), \(E\), \(F\) are collinear, by the definition of a straight - angle, \(\angle CEF\) is a straight angle. So \(m\angle CEF = 180^{\circ}\).

Step2: Analyze the right - angle property

Given that \(\overrightarrow{EA}\perp\overrightarrow{EC}\) (implied by the right - angle mark at \(E\) between \(A\) and \(C\)), by the definition of perpendicular lines, \(\angle AEC = 90^{\circ}\). Also, \(\angle AEF=\angle AEC+\angle CEF - \angle AEC\) (since \(\angle CEF = 180^{\circ}\)), so \(\angle AEF\) is a right angle (\(m\angle AEF=90^{\circ}\)).

Step3: Check other options

• For \(m\angle CEA = 90^{\circ}\): There is no information from the angle - bisector property (only that \(\overrightarrow{EB}\) bisects \(\angle CEA\)) to suggest \(m\angle CEA = 90^{\circ}\).
• For \(m\angle CEF=m\angle CEA + m\angle BEF\): This is incorrect. If \(\overrightarrow{EB}\) bisects \(\angle CEA\), let \(\angle CEB=\angle BEA = x\). But \(\angle CEF\) is a straight angle (\(180^{\circ}\)) and has no such additive relationship with \(\angle CEA\) and \(\angle BEF\) as given.
• For \(m\angle CEB = 2(m\angle CEA)\): If \(\overrightarrow{EB}\) bisects \(\angle CEA\), then \(m\angle CEA=2m\angle CEB\), so this option is wrong.

Answer:

\(\angle CEF\) is a straight angle, \(\angle AEF\) is a right angle.