Question

ray ce is the angle bisector of ∠acd. which statement about the figure must be true?
○ m∠ecd = 1/2m∠ecb
○ m∠ace = 1/2m∠acd
○ ∠ace ≅ ∠dcb
○ ∠ecd ≅ ∠acd

Explanation:

Step1: Recall angle - bisector definition

An angle - bisector divides an angle into two equal angles. Since ray $CE$ is the angle - bisector of $\angle ACD$, we know that $\angle ACE$ and $\angle ECD$ are equal, and each is half of $\angle ACD$. That is, $m\angle ACE=\frac{1}{2}m\angle ACD$ and $m\angle ECD = \frac{1}{2}m\angle ACD$.

Step2: Analyze each option

• Option 1: There is no information to suggest that $m\angle ECD=\frac{1}{2}m\angle ECB$. $\angle ECD$ and $\angle ECB$ have no such defined relationship based on the given that $CE$ bisects $\angle ACD$.
• Option 2: By the definition of an angle - bisector, if $CE$ bisects $\angle ACD$, then $m\angle ACE=\frac{1}{2}m\angle ACD$. This is correct.
• Option 3: $\angle ACE$ and $\angle DCB$ are not necessarily congruent. There is no information in the problem to support this equality.
• Option 4: $\angle ECD$ is half of $\angle ACD$, so $\angle ECD

eq\angle ACD$.

Answer:

$m\angle ACE=\frac{1}{2}m\angle ACD$ (the option that reads $m\angle ACE=\frac{1}{2}m\angle ACD$)