Question

vi. jessica wrote the following 2 - column proof. what reason is jessica missing? diagram: two triangles, top (bca) with ∠b=30°, ∠c=80°, bc=7cm; bottom (efd) with ∠e=30°, ∠f=80°, ef=7cm. two - column proof: statement: ∠b ≅ ∠e, reason: given; statement: bc ≅ ef, reason: given; statement: ∠c ≅ ∠f, reason: given; statement: δbca ≅ δefd, reason: ? options: asa congruence postulate, aas congruence postulate, aaa congruence postulate, sss congruence postulate.

Explanation:

To determine the missing reason for the congruence of \(\triangle BCA\) and \(\triangle EFD\), we analyze the given information:

• \(\angle B \cong \angle E\) (given),
• \(BC \cong EF\) (given),
• \(\angle C \cong \angle F\) (given).

We check the triangle congruence postulates:

• ASA (Angle - Side - Angle): Requires two angles and the included side to be congruent. Here, \(\angle B\) and \(\angle C\) are angles with the included side \(BC\) in \(\triangle BCA\), and \(\angle E\) and \(\angle F\) are angles with the included side \(EF\) in \(\triangle EFD\). Since \(\angle B\cong\angle E\), \(BC\cong EF\), and \(\angle C\cong\angle F\), this satisfies the ASA postulate.
• AAS (Angle - Angle - Side): Requires two angles and a non - included side. But in our case, the side \(BC\) (or \(EF\)) is included between the two angles, so AAS is not applicable.
• AAA (Angle - Angle - Angle): Only shows similarity, not congruence, so it's incorrect.
• SSS (Side - Side - Side): Requires three sides to be congruent, but we only have one side given as congruent, so it's incorrect.

Answer:

A. ASA Congruence Postulate