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
2 | \\(\angle aeb\\) is a right angle | perpendicular lines form right angles
(there is a diagram of a quadrilateral with points b, c, e, d, a, and a right angle at e between ac and bd.)

Explanation:

Step1: Identify Midpoint Property

Since \( E \) is the midpoint of \( \overline{BD} \), by the definition of midpoint, \( \overline{BE} \cong \overline{DE} \).

Step2: Identify Common Side

\( \overline{AE} \) is a common side to both \( \triangle BAE \) and \( \triangle DAE \), so \( \overline{AE} \cong \overline{AE} \) (Reflexive Property of Congruence).

Step3: Identify Right Angles

From \( \overline{AC} \perp \overline{BD} \), \( \angle AEB \) and \( \angle AED \) are right angles, so \( \angle AEB \cong \angle AED \) (All right angles are congruent).

Step4: Apply SAS Congruence

Now we have \( \overline{BE} \cong \overline{DE} \), \( \angle AEB \cong \angle AED \), and \( \overline{AE} \cong \overline{AE} \). By the Side - Angle - Side (SAS) Congruence Postulate, \( \triangle BAE \cong \triangle DAE \).

Answer:

By showing \( \overline{BE} \cong \overline{DE} \) (midpoint), \( \angle AEB \cong \angle AED \) (right angles), and \( \overline{AE} \cong \overline{AE} \) (reflexive), we use SAS to prove \( \triangle BAE \cong \triangle DAE \).