Question

given that ∠cea is a right angle and ray bisects ∠cea, which statement must be true?
○ ∠bea ≅ ∠cea
○ ∠ceb ≅ ∠cea
○ ( mangle ceb = 45^circ )
○ ( mangle cea = 45^circ )

Explanation:

1. First, recall that a right angle (∠CEA) measures \( 90^\circ \) because \( \angle CEA \) is a right angle.
2. Then, since \( \overrightarrow{EB} \) bisects \( \angle CEA \), it divides \( \angle CEA \) into two equal angles. By the definition of an angle bisector, \( \angle CEB=\angle BEA \), and each of these angles is half of \( \angle CEA \).
3. Calculate the measure of \( \angle CEB \): \( m\angle CEB=\frac{1}{2}m\angle CEA \). Substituting \( m\angle CEA = 90^\circ \), we get \( m\angle CEB=\frac{90^\circ}{2}=45^\circ \).
4. Now, analyze the other options:

• \( \angle BEA\cong\angle CEA \): \( \angle BEA \) is half of \( \angle CEA \) (which is \( 90^\circ \)), so \( \angle BEA = 45^\circ \), and \( \angle CEA=90^\circ \). They are not congruent.
• \( \angle CEB\cong\angle CEA \): \( \angle CEB = 45^\circ \) and \( \angle CEA = 90^\circ \), so they are not congruent.
• \( m\angle CEA = 45^\circ \): But \( \angle CEA \) is a right angle, so its measure is \( 90^\circ \), not \( 45^\circ \).

Answer:

\( m\angle CEB = 45^\circ \) (the third option)