Question

1. alyssa has written the following proof.

given: e is the midpoint of $overline{ac}$. $overline{ab} \parallel \overline{dc}$
prove: $\triangle aeb \cong \triangle ced$
proof:
statements | reasons

1. e is the midpoint of $overline{ac}$. | 1. given
2. $overline{ab} \parallel \overline{dc}$ | 2. given
3. $overline{ae} \cong \overline{ec}$ | 3. definition of midpoint
4. $\angle aeb \cong \angle ced$ | 4. vertical angles are congruent.
5. $\angle eab \cong \angle ecd$ | 5. if two parallel lines are intersected by a transversal, then alternate interior angles are congruent.
6. $\triangle aeb \cong \triangle ced$ | 6. blank

which of the following would correctly complete statement 6?
options: asa, aas, sss, sas

Explanation:

To determine the congruence criterion for \(\triangle AEB \cong \triangle CED\), we analyze the given information:

• From Statement 3, \(AE = EC\) (E is the midpoint of \(AC\)).
• From Statement 4, \(\angle AEB=\angle CED\) (vertical angles are congruent).
• From Statement 5, \(\angle EAB=\angle ECD\) (alternate interior angles, since \(AB \parallel DC\)).

We have two angles and the included side congruent: \(\angle EAB=\angle ECD\), \(AE = EC\), and \(\angle AEB=\angle CED\). This matches the ASA (Angle - Side - Angle) congruence criterion, where two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle.

Answer:

ASA (Angle - Side - Angle)