Question

given: e is the midpoint of \\(\overline{bd}\\) and \\(\overline{ac} \perp \overline{bd}\\).
prove: \\(\triangle bae \cong \triangle dae\\).
step | statement | reason
1 | e is the midpoint of \\(\overline{bd}\\) \\(\overline{ac} \perp \overline{bd}\\) | given
try | \\(\angle \square\\) is a right angle | select a reason...
reasons: perpendicular lines form right angles; an angle supplementary to a right angle is a right angle; angles which are supplementary and congruent are right angles; an angle congruent to a right angle is a right angle

Explanation:

Step1: Analyze the Given Information

We know that \(\overline{AC} \perp \overline{BD}\). By the definition of perpendicular lines, when two lines are perpendicular, they form right angles at their intersection. The intersection of \(\overline{AC}\) and \(\overline{BD}\) is point \(E\) (since \(E\) is on both \(\overline{BD}\) and \(\overline{AC}\) as \(E\) is the midpoint of \(\overline{BD}\) and \(\overline{AC}\) is perpendicular to \(\overline{BD}\)). So the angles \(\angle BAE\) and \(\angle DAE\) are right angles. The reason for \(\angle BAE\) (or \(\angle DAE\)) being a right angle is "Perpendicular lines form right angles".

Step2: Identify the Correct Reason

From the given options, the reason that fits is "Perpendicular lines form right angles" because we have \(\overline{AC} \perp \overline{BD}\), so the angles formed at their intersection (at \(E\)) are right angles.

Answer:

The correct reason is "Perpendicular lines form right angles" and the angle (for example, \(\angle BAE\) or \(\angle DAE\)) is a right angle. So the reason to select is "Perpendicular lines form right angles".