Question

for the given picture, what additional piece of information would be needed to prove triangle bac is congruent to triangle dac by sas?
options:

1. ( mangle bda = mangle dca )
2. ( mangle bac = mangle dac )
3. ( mangle abc = mangle adc )
4. ( ac = ac )

Explanation:

Step1: Recall SAS Congruence

SAS (Side - Angle - Side) congruence criterion 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 Given Information

From the diagram, we can see that \(AB = AD\) (marked with the same tick marks) and \(AC\) is a common side, so \(AC=AC\). For SAS, we need the included angle between these two sides to be congruent. The included angle for \(\triangle BAC\) is \(\angle BAC\) and for \(\triangle DAC\) is \(\angle DAC\). So we need \(m\angle BAC=m\angle DAC\).

Step3: Evaluate Other Options

• Option \(AC = AC\) is already a common side, not additional info.
• \(m\angle BDA=m\angle DCA\) and \(m\angle ABC = m\angle ADC\) are not related to the included angle for SAS.

Answer:

\(m\angle BAC = m\angle DAC\)