Question

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

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal - measure angles. If \(CE\) is the angle bisector of \(\angle ACD\), then \(m\angle ACE=m\angle ECD\) and \(m\angle ACE=\frac{1}{2}m\angle ACD\), \(m\angle ECD = \frac{1}{2}m\angle ACD\).

Step2: Analyze each option

• Option 1:

\(m\angle ECD=\frac{1}{2}m\angle ECB\) is not correct. There is no information from the angle - bisector definition to support this relationship.

• Option 2:

Since \(CE\) bisects \(\angle ACD\), by the definition of an angle bisector \(m\angle ACE=\frac{1}{2}m\angle ACD\).

• Option 3:

\(\angle ACE\cong\angle DCB\) is not correct. There is no reason to assume these two angles are congruent from the given information that \(CE\) is the bisector of \(\angle ACD\).

• Option 4:

\(\angle ECD\cong\angle ACD\) is not correct. Since \(m\angle ECD=\frac{1}{2}m\angle ACD\) (from the angle - bisector definition), they are not congruent (unless \(m\angle ACD = 0^{\circ}\), which is not the case for non - degenerate angles in a geometric figure like this).

Answer:

\(m\angle ACE=\frac{1}{2}m\angle ACD\)