Question

the diagram shows that <dac = <bca and ad = cb. what additional information is needed to show the two triangles are congruent by asa? 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 two angles and the included side. We already have one angle ($\angle DAC=\angle BCA$) and one side ($AD = CB$). We need another pair of angles such that the side $AC$ is included between the two angles.

Looking at the angles, if $\angle DCA=\angle BAC$, then in $\triangle DAC$ and $\triangle BCA$:

• $\angle DAC=\angle BCA$ (given)
• $AC = CA$ (common side)
• $\angle DCA=\angle BAC$ (the additional angle we need)

This would satisfy the ASA criterion as the side $AC$ is included between $\angle DAC$ and $\angle DCA$ in $\triangle DAC$ and between $\angle BCA$ and $\angle BAC$ in $\triangle BCA$.

Let's analyze the other options:

• Option 1: $AB = CD$: This would be related to SSS or SAS, not ASA.
• Option 3: $AC = CA$: This is a common side, but we already know the side $AC$ is common. We need an angle, not just the side.
• Option 4: $\angle B=\angle D$: This would be related to AAS, not ASA.

Step2: Confirm the Correct Option

Based on the above analysis, the additional information needed to show congruence by ASA is $\angle DCA=\angle BAC$.

Answer:

$\boldsymbol{\angle DCA=\angle BAC}$ (the option with $\angle DCA=\angle BAC$)