Question

ray ce is the angle bisector of $angle acd$. which statement about the figure must be true?
$\bigcirc$ $mangle ecd = \frac{1}{2}m\angle ecb$
$\bigcirc$ $m\angle ace = \frac{1}{2}m\angle acd$
$\bigcirc$ $\angle ace \cong \angle dcb$
$\bigcirc$ $\angle ecd \cong \angle acd$

Explanation:

Step1: Recall angle bisector definition

An angle bisector splits an angle into two congruent angles, each equal to half the original angle. For $\angle ACD$, ray $CE$ divides it into $\angle ACE$ and $\angle ECD$, so $m\angle ACE = m\angle ECD = \frac{1}{2}m\angle ACD$.

Step2: Evaluate each option

• Option1: $m\angle ECD = \frac{1}{2}m\angle ECB$: No basis, $\angle ECB$ is not related to the bisected angle.
• Option2: $m\angle ACE = \frac{1}{2}m\angle ACD$: Matches the angle bisector property.
• Option3: $\angle ACE \cong \angle DCB$: No given relation between these angles.
• Option4: $\angle ECD \cong \angle ACD$: $\angle ECD$ is half of $\angle ACD$, not congruent.

Answer:

B. $m\angle ACE = \frac{1}{2}m\angle ACD$