Question

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

Explanation:

Step1: Recall the definition of an angle bisector

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

Step2: Analyze each option

• For the option \(m\angle ECD=\frac{1}{2}m\angle ECB\): Since \(CE\) is the bisector of \(\angle ACD\), not related to \(\angle ECB\) in this way.
• For the option \(m\angle ACE = \frac{1}{2}m\angle ACD\): By the definition of an angle bisector, if \(CE\) bisects \(\angle ACD\), then \(\angle ACE\) and \(\angle ECD\) are the two equal - parts of \(\angle ACD\). So \(m\angle ACE=\frac{1}{2}m\angle ACD\) (and \(m\angle ECD=\frac{1}{2}m\angle ACD\)).
• For the option \(\angle ACE\cong\angle DCB\): There is no information given to suggest that these two angles are congruent.
• For the option \(\angle ECD\cong\angle ACD\): Since \(\angle ECD=\frac{1}{2}\angle ACD\) (from the angle - bisector definition), they are not congruent.

Answer:

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