Question

jeff drew the figure shown. given: ∠a ≅ ∠d. choose all conditions that can be independently used to conclude that △abe ≅ △bcd. ∠e ≅ ∠c, (overline{cd} cong overline{db}), (overline{eb} cong overline{cb}), (overline{ae} cong overline{eb}), (overline{ab} cong overline{db})

Explanation:

To determine which conditions can be used to conclude \(\triangle ABE \cong \triangle BCD\) given \(\angle A \cong \angle D\), we analyze each option using triangle congruence criteria (ASA, AAS, SAS, SSS).

Step 1: Analyze \(\angle E \cong \angle C\)

If \(\angle A \cong \angle D\) (given) and \(\angle E \cong \angle C\), we also note that \(\angle ABE\) and \(\angle DBC\) are vertical angles, so \(\angle ABE \cong \angle DBC\). By AAS (Angle - Angle - Side), \(\triangle ABE \cong \triangle BCD\) (since two angles and a non - included side are congruent).

Step 2: Analyze \(\overline{CD} \cong \overline{DB}\)

This gives a side in \(\triangle BCD\), but there is no clear way to relate this side to \(\triangle ABE\) with the given \(\angle A\cong\angle D\) to prove congruence. It does not fit any of the standard triangle congruence criteria (ASA, AAS, SAS, SSS) for these two triangles.

Step 3: Analyze \(\overline{EB} \cong \overline{CB}\)

We know \(\angle A\cong\angle D\) and \(\angle ABE\cong\angle DBC\) (vertical angles). If \(\overline{EB} \cong \overline{CB}\), by AAS (Angle - Angle - Side), we can conclude \(\triangle ABE\cong\triangle BCD\) (two angles and a non - included side: \(\angle A\cong\angle D\), \(\angle ABE\cong\angle DBC\), \(\overline{EB} \cong \overline{CB}\)).

Step 4: Analyze \(\overline{AE} \cong \overline{EB}\)

This is a side within \(\triangle ABE\) and does not provide a relationship between the sides or angles of \(\triangle ABE\) and \(\triangle BCD\) that can be used with \(\angle A\cong\angle D\) to prove congruence. It does not satisfy any triangle congruence criterion for the two triangles.

Step 5: Analyze \(\overline{AB} \cong \overline{DB}\)

We know \(\angle A\cong\angle D\) and \(\angle ABE\cong\angle DBC\) (vertical angles). If \(\overline{AB} \cong \overline{DB}\), by ASA (Angle - Side - Angle), we can conclude \(\triangle ABE\cong\triangle BCD\) (two angles and the included side: \(\angle A\cong\angle D\), \(\overline{AB} \cong \overline{DB}\), \(\angle ABE\cong\angle DBC\)).

Answer:

The conditions that can be independently used to conclude \(\triangle ABE\cong\triangle BCD\) are:

• \(\boldsymbol{\angle E \cong \angle C}\)
• \(\boldsymbol{\overline{EB} \cong \overline{CB}}\)
• \(\boldsymbol{\overline{AB} \cong \overline{DB}}\)