Question

given that \\( \angle cea \\) is a right angle and \\( \overrightarrow{eb} \\) bisects \\( \angle cea \\), which statement must be true? \\( \angle bea \cong \angle cea \\) \\( \angle ceb \cong \angle cea \\) \\( m \angle ceb = 45 ^ { \circ } \\) \\( m \angle cea = 45 ^ { \circ } \\)

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal - measure angles.

Step2: Use the given information about the right - angle

Since \(\angle CEA = 90^{\circ}\) and \(\overrightarrow{EB}\) bisects \(\angle CEA\), then \(\angle CEB=\angle BEA=\frac{1}{2}\angle CEA\).

Step3: Calculate the measure of \(\angle CEB\)

Substitute \(\angle CEA = 90^{\circ}\) into the formula \(\angle CEB=\frac{1}{2}\angle CEA\). So, \(\angle CEB=\frac{1}{2}\times90^{\circ}=45^{\circ}\).

Now let's check each option:

• For \(\angle BEA\cong\angle CEA\): Since \(\angle BEA = 45^{\circ}\) and \(\angle CEA=90^{\circ}\), \(45^{\circ}

eq90^{\circ}\), so this option is false.

• For \(\angle CEB\cong\angle CEA\): Since \(\angle CEB = 45^{\circ}\) and \(\angle CEA = 90^{\circ}\), \(45^{\circ}

eq90^{\circ}\), so this option is false.

• For \(m\angle CEB = 45^{\circ}\): As we calculated above, this is true.
• For \(m\angle CEA=45^{\circ}\): Given that \(\angle CEA\) is a right - angle, \(m\angle CEA = 90^{\circ}

eq45^{\circ}\), so this option is false.

Answer:

\(m\angle CEB = 45^{\circ}\)