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

Let's analyze the triangles $\triangle DAC$ and $\triangle BCA$:

• In $\triangle DAC$, the angles are $\angle DAC$, $\angle DCA$ and the included side between them is $AC$.
• In $\triangle BCA$, the angles are $\angle BCA$, $\angle BAC$ and the included side between them is $AC$.

We already have $\angle DAC=\angle BCA$. If we have $\angle DCA=\angle BAC$, then:

• For $\triangle DAC$: Angles $\angle DAC$ and $\angle DCA$ with included side $AC$.
• For $\triangle BCA$: Angles $\angle BCA$ and $\angle BAC$ with included side $AC$.

Now let's analyze the other options:

• Option $AB = CD$: This is a side - side - side or side - angle - side related, not ASA.
• Option $AC=CA$: This is the reflexive property, but it's the common side, we need another angle for ASA.
• Option $\angle B=\angle D$: This is related to AAS or other congruence, not ASA.

Answer:

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