Question

the diagram shows that <dac = <bca and ad = cb. what additional information is needed to show the two triangles are congruent by asa? * 4 points options: ab = cd, <dca = <bac, ac = ca, <b = <d

Explanation:

Step1: Recall ASA Congruence

ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

We know that $\angle DAC=\angle BCA$ and $AD = CB$. Let's consider triangles $\triangle DAC$ and $\triangle BCA$. The side $AC$ is common to both triangles. For ASA, we need another pair of angles such that the side $AC$ is included between the two angles.

Step2: Analyze Each Option

• Option 1: $AB = CD$: This gives a side - side - side or side - angle - side - like situation, not related to ASA directly as it doesn't provide the required angle.
• Option 2: $\angle DCA=\angle BAC$: In $\triangle DAC$ and $\triangle BCA$, we have $\angle DAC=\angle BCA$ (given), $AC = CA$ (common side), and if $\angle DCA=\angle BAC$, then we have two angles and the included side (the included side for $\angle DAC$ and $\angle DCA$ is $AC$, and for $\angle BCA$ and $\angle BAC$ is $AC$) which satisfies ASA.
• Option 3: $AC = CA$: This is the reflexive property (common side), but we already know a side and one angle, we need another angle for ASA, not just the common side re - stated.
• Option 4: $\angle B=\angle D$: This is not related to the angles required for ASA as it doesn't involve the angles around the included side.

Answer:

$\angle DCA=\angle BAC$ (the option with $\angle DCA = \angle BAC$)