Question

1. what other information do you need in order to prove the triangles congruent using the sas congruence postulate? a ( angle b a c=angle d a c ) b ( overline{a b}=overline{a d} ) c ( angle c b a=angle c d a ) d ( overline{a b} perp overline{a d} ) 4. which triangles are congruent by asa?

Explanation:

Step1: Recall the SAS Congruence Postulate

The SAS (Side - Angle - Side) Congruence 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.
In the given figure, we have \(AC = AC\) (common side). For \(\triangle ABC\) and \(\triangle ADC\), if we want to use SAS, we need to have the included angles between the sides \(AB\) and \(AC\), \(AD\) and \(AC\) congruent.

Step2: Analyze each option

• Option A: \(\angle BAC\cong\angle DAC\). If \(\angle BAC\cong\angle DAC\), \(AC = AC\) (common side), and assume \(AB = AD\) (if we consider the sides adjacent to the angles). But actually, if we have \(AC\) as a common side, and \(\angle BAC\cong\angle DAC\), and if we assume \(BC = DC\) (from the figure's structure, \(C\) is a point on \(BD\)), by SAS (\(AC = AC\), \(\angle BAC=\angle DAC\), \(AB = AD\) (if we consider the two - side - angle structure for \(\triangle ABC\) and \(\triangle ADC\))).
• Option B: \(\overline{AB}\cong\overline{AD}\) only gives one side, we still need the included angle.
• Option C: \(\angle CBA\cong\angle CDA\) is not the included angle for the SAS postulate.
• Option D: \(\overline{AB}\perp\overline{AD}\) gives a right - angle information, but not relevant for the SAS postulate in the context of proving \(\triangle ABC\) and \(\triangle ADC\) congruent.

Answer:

A. \(\angle BAC\cong\angle DAC\)