Question

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

Explanation:

1. From the diagram, \( \angle CEA \) is a right angle (marked with a square), so \( m\angle CEA = 90^\circ \), this statement is true.
2. \( \angle CEF \) is a straight angle (since \( C - E - F \) are colinear), and \( \angle CEF=m\angle CEA + m\angle AEF \), but also \( \angle AEF = 90^\circ\) (as \( \angle CEA = 90^\circ\) and \( \angle CEA+\angle AEF = 180^\circ\)), and since \( \overrightarrow{EB}\) bisects \( \angle CEA \), \( \angle CEB=\angle BEA = 45^\circ\), but \( \angle CEF=m\angle CEA + m\angle BEF\) is incorrect. Wait, actually, \( \angle CEF\) is a straight angle (\(180^\circ\)), \( \angle CEA = 90^\circ\), \( \angle AEF=90^\circ\) (so \( \angle AEF\) is a right angle). Also, \( \angle CEF\) is a straight angle (since \( C, E, F\) are on a straight line). Let's re - evaluate:

• \( m\angle CEA = 90^\circ\): The diagram shows a right - angle symbol at \( \angle CEA\), so this is true.
• \( m\angle CEF=m\angle CEA + m\angle BEF\): \( \angle CEF=\angle CEB+\angle BEF\), and since \( \overrightarrow{EB}\) bisects \( \angle CEA\) (\( \angle CEB=\angle BEA = 45^\circ\)) and \( \angle CEA = 90^\circ\), \( \angle CEF = 180^\circ\) (straight angle), \( \angle CEA=90^\circ\), \( \angle BEF=\angle BEA+\angle AEF=45^\circ + 90^\circ=135^\circ\), \( \angle CEA+\angle BEF=90^\circ + 135^\circ = 225^\circ

eq180^\circ\), so this is false.

• \( m\angle CEB = 2(m\angle CEA)\): \( m\angle CEB = 45^\circ\), \( 2(m\angle CEA)=180^\circ\), so this is false.
• \( \angle CEF\) is a straight angle: Since points \( C\), \( E\), and \( F\) are colinear (the line \( CF\) is a straight line), \( \angle CEF = 180^\circ\), so it is a straight angle, this is true.
• \( \angle AEF\) is a right angle: \( \angle CEA = 90^\circ\) and \( \angle CEA+\angle AEF = 180^\circ\) (linear pair), so \( \angle AEF=180^\circ - 90^\circ = 90^\circ\), so \( \angle AEF\) is a right angle, this is true.

Answer:

• \( m\angle CEA = 90^\circ\)
• \( \angle CEF\) is a straight angle.
• \( \angle AEF\) is a right angle.