Question

1. what additional information is needed to prove the triangles are congruent by the sas postulate?

∠cba ≅ ∠cda

ab ≅ ad

ab ⊥ ad

∠bac ≅ ∠dac

Explanation:

Step1: Recall the SAS Postulate

The SAS (Side - Angle - Side) Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze the given triangles

In \(\triangle ABC\) and \(\triangle ADC\), we already know that \(\angle BAC\cong\angle DAC\) (given as the angle part of SAS) and \(AC = AC\) (common side). For SAS, we need the other pair of sides adjacent to the given angle.

Step3: Check each option

• Option \(\angle CBA\cong\angle CDA\): This is an angle - angle - side (AAS) or angle - side - angle (ASA) related information, not SAS.
• Option \(\overline{AB}\cong\overline{AD}\): If \(\overline{AB}\cong\overline{AD}\), in \(\triangle ABC\) and \(\triangle ADC\), we have \(\overline{AB}\cong\overline{AD}\), \(\angle BAC\cong\angle DAC\), and \(\overline{AC}\cong\overline{AC}\). By the SAS Postulate (\(AB = AD\), \(\angle BAC=\angle DAC\), \(AC = AC\)), \(\triangle ABC\cong\triangle ADC\).
• Option \(\overline{AB}\perp\overline{AD}\): This gives information about the angle between \(AB\) and \(AD\), not relevant for the SAS of \(\triangle ABC\) and \(\triangle ADC\).
• Option \(\angle BAC\cong\angle DAC\): This is already given (not an additional information for the purpose of finding what is needed as per the question's requirement of "additional information").

Answer:

\(\overline{AB}\cong\overline{AD}\)