Question

point b is the midpoint of \\( \overline { a c } \\).
which statements about the figure must be true?
select three options.
\\( \square \angle d b c \\) is bisected by ray bd.
\\( \square \angle a b c \\) is bisected by ray bd.
\\( \square b c = \frac { 1 } { 2 } a c \\)
\\( \square \overline { d b } \cong \overline { b c } \\)
\\( \square 2 m \angle d b c = m \angle a b c \\)

Explanation:

Step1: Analyze the mid - point property

Since \(B\) is the mid - point of \(\overline{AC}\), by the definition of a mid - point, \(BC=\frac{1}{2}AC\).

Step2: Analyze the angle bisector

\(\angle ABC = 180^{\circ}\) (a straight angle). \(\angle ABD=\angle DBC = 90^{\circ}\) (because \(BD\perp AC\)). So, ray \(BD\) bisects \(\angle ABC\) (\(\angle ABD=\angle DBC\)).

Step3: Analyze the angle relationship

Since \(\angle ABC = 180^{\circ}\) and \(\angle DBC=90^{\circ}\), then \(2m\angle DBC=m\angle ABC\) (\(2\times90^{\circ}=180^{\circ}\)).

Answer:

\(\angle ABC\) is bisected by ray \(BD\), \(BC = \frac{1}{2}AC\), \(2m\angle DBC=m\angle ABC\)