Question

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

Explanation:

Step1: Analyze the angle bisector

Since \(\overrightarrow{EB}\) bisects \(\angle CEA\), we know that \(m\angle CEB = m\angle BEA\), and \(m\angle CEA=m\angle CEB + m\angle BEA = 2m\angle CEB\) (or \(2m\angle BEA\)). Also, from the diagram, \(\angle AEF\) and \(\angle FEC\) related? Wait, the right angle: \(\angle AEC\) looks like a right angle (the red square), so \(m\angle CEA = 90^\circ\). Then:

1. \(m\angle CEA = 90^\circ\): From the diagram, \(\angle AEC\) has a right - angle symbol, so this is true.
2. \(\angle AEF\) is a right angle: Since \(\angle FEC\) is a straight line? Wait, \(F - E - C\) is a straight line, so \(\angle FEC = 180^\circ\). \(\angle AEC=90^\circ\), so \(\angle AEF=\angle FEC-\angle AEC = 180^\circ - 90^\circ=90^\circ\), so \(\angle AEF\) is a right angle, this is true.
3. \(\angle CEF\) is a straight angle: \(F\), \(E\), \(C\) are colinear (on a straight line), so \(\angle CEF = 180^\circ\), which is a straight angle, this is true.
4. \(m\angle CEF=m\angle CEA + m\angle BEF\): \(\angle CEF = 180^\circ\), \(\angle CEA = 90^\circ\), \(\angle BEF=\angle AEF+\angle AEB\). Since \(\angle AEF = 90^\circ\) and \(\angle AEB=\frac{1}{2}\angle CEA = 45^\circ\) (because \(EB\) bisects \(\angle CEA\)), \(\angle BEF=90^\circ + 45^\circ = 135^\circ\), \(\angle CEA+\angle BEF=90^\circ+135^\circ = 225^\circ

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

1. \(m\angle CEB = 2(m\angle CEA)\): \(m\angle CEA = 90^\circ\), \(m\angle CEB=\frac{1}{2}m\angle CEA = 45^\circ\), \(2(m\angle CEA)=180^\circ

eq45^\circ\), so this is false.

Answer:

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