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

Explanation:

Step1: Identify Midpoint Property

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

Step2: Identify Right Angles

Given \( \overline{AC} \perp \overline{BD} \), by the definition of perpendicular lines, \( \angle BEA \) and \( \angle DEA \) are right angles, so \( \angle BEA \cong \angle DEA \) (all right angles are congruent).

Step3: 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).

Step4: Apply SAS Congruence

By the Side - Angle - Side (SAS) congruence postulate, since \( \overline{BE} \cong \overline{DE} \), \( \angle BEA \cong \angle DEA \), and \( \overline{AE} \cong \overline{AE} \), we can conclude that \( \triangle BAE \cong \triangle DAE \).

Answer:

To prove \( \triangle BAE \cong \triangle DAE \):

1. \( \overline{BE} \cong \overline{DE} \) (Definition of midpoint, as \( E \) is midpoint of \( \overline{BD} \)).
2. \( \angle BEA \cong \angle DEA \) (Definition of perpendicular lines, as \( \overline{AC} \perp \overline{BD} \), so both are right angles and right angles are congruent).
3. \( \overline{AE} \cong \overline{AE} \) (Reflexive property of congruence).
4. \( \triangle BAE \cong \triangle DAE \) (SAS Congruence Postulate, using the three statements above).