Question

given: e is the midpoint of \\(\overline{bd}\\) and \\(\overline{ac} \perp \overline{bd}\\). prove: \\(\triangle bae \cong \triangle dae\\). step 1: statement - e is the midpoint of \\(\overline{bd}\\), \\(\overline{ac} \perp \overline{bd}\\); reason - given. step 2: statement - \\(\angle aeb\\) is a right angle; reason - perpendicular lines form right angles. then there is a statement \\(\overline{be} \cong \overline{de}\\) with a dropdown to select a reason, including options like reflexive property, transitive property, substitution, in a triangle, sides opposite of congruent angles are congruent.

Explanation:

Step1: Recall Midpoint Definition

A midpoint of a segment divides it into two congruent segments. Since \( E \) is the midpoint of \( \overline{BD} \), by the definition of a midpoint, \( \overline{BE} \cong \overline{DE} \).

Step2: Match with Given Options

The reason for \( \overline{BE} \cong \overline{DE} \) is the definition of a midpoint (though in the dropdown, we need to check, but the key is the midpoint divides the segment into two congruent parts. Wait, the dropdown options—wait, maybe the initial thought: the midpoint creates two equal segments, so the reason here, but the dropdown has options. Wait, the correct reason for \( BE \cong DE \) when \( E \) is the midpoint of \( BD \) is "Definition of Midpoint" (not in the shown dropdown? Wait, the shown dropdown has Reflexive, Transitive, Substitution, etc. Wait, maybe a mistake, but the correct reason for \( BE \cong DE \) is that a midpoint divides a segment into two congruent segments. But among the given options, maybe the intended is that (but the options given: Reflexive is for a segment congruent to itself, Transitive is if \( a\cong b \) and \( b\cong c \) then \( a\cong c \), Substitution is replacing. Wait, no—wait, the midpoint definition: if \( E \) is the midpoint of \( BD \), then \( BE = DE \), so \( \overline{BE} \cong \overline{DE} \) by Definition of Midpoint. But since that's not in the dropdown, maybe the problem has a typo, but the correct reason here is related to midpoint. However, among the given options, maybe the closest is—wait, no, the options are wrong? Wait, no, maybe I misread. Wait, the step is \( BE \cong DE \), reason: since \( E \) is midpoint, so \( BE = DE \), so congruent. So the reason should be "Definition of Midpoint", but since it's not in the dropdown, maybe the problem's dropdown is missing, but the correct answer here is that the reason for \( BE \cong DE \) is "Definition of Midpoint", but among the given options, maybe the intended is—wait, no, the user is to select the reason. Wait, the options are Reflexive Property, Transitive Property, Substitution, and "In a triangle, sides opposite...". Wait, no, the correct reason is Definition of Midpoint, but since it's not there, maybe the problem has an error. But assuming that the options are as given, maybe the intended is—wait, no, the midpoint divides the segment into two equal parts, so \( BE \cong DE \) by Definition of Midpoint. So the answer is that the reason for \( BE \cong DE \) is "Definition of Midpoint" (even if not in the dropdown, but maybe the dropdown is incomplete). But the question is to select the reason, so the correct option (if Definition of Midpoint was there) but since it's not, maybe the problem is wrong. But according to the problem, the correct reason for \( BE \cong DE \) when \( E \) is the midpoint of \( BD \) is "Definition of Midpoint".

Answer:

The correct reason for \( \overline{BE} \cong \overline{DE} \) is "Definition of Midpoint" (note: if this is not in the dropdown, there may be a problem with the given options, but the logical reason is Definition of Midpoint as \( E \) being the midpoint of \( \overline{BD} \) implies \( \overline{BE} \cong \overline{DE} \)).