adjust point d so the measure of angle bad is equal to the measure of angle cad. which statements are true? check all that apply. $\overline{ad}$ bisects $\angle bac$. $\overline{ad}$ bisects $\overline{bc}$. $\overline{ad}$ forms right angles with $\overline{bc}$. $\overline{ad}$ is perpendicular to $\overline{bc}$. $\overline{ad}$ is the perpendicular bisector of $\overline{bc}$. $m\angle bad = 20^{\circ}$ $m\angle cad = 54^{\circ}$ $bd = 18$ units $dc = 42$ units

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# Brief Explanations:
1. For the first statement: By definition, an angle bisector divides an angle into two equal angles. Since we adjust $ D $ so $ m\angle BAD = m\angle CAD $, $ \overline{AD} $ bisects $ \angle BAC $. So this statement is true.
2. For the second statement: A segment bisector of $ \overline{BC} $ would mean $ BD = DC $. But here $ BD = 18 $ units and $ DC = 42 $ units, so $ \overline{AD} $ does not bisect $ \overline{BC} $. This statement is false.
3. For the third statement: There is no information given (and from the angle measures $ 20^\circ $ and $ 54^\circ $ we can infer it's not a right angle situation) that $ \overline{AD} $ forms right angles with $ \overline{BC} $. This statement is false.
4. For the fourth statement: Perpendicular means forming a right angle ($ 90^\circ $). Since the angles $ \angle BAD $ and $ \angle CAD $ are $ 20^\circ $ and $ 54^\circ $ (after adjustment, but even if they were equal, there's no indication of right angle with $ \overline{BC} $), $ \overline{AD} $ is not perpendicular to $ \overline{BC} $. This statement is false.
5. For the fifth statement: A perpendicular bisector needs to be both perpendicular to $ \overline{BC} $ and bisect $ \overline{BC} $. We already saw $ BD
eq DC $ and no right angle, so this is false.

# Answer:
$\boldsymbol{\overline{AD}}$ bisects $\boldsymbol{\angle BAC}$ (the first option).

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