Question

the diagram shows that <dac = <bca and ad = cb. what additional information is needed to show the two triangles are congruent by sas? ab = cd; <cda = <acb; ac = ca; <b = <d

Explanation:

Step1: Recall SAS Congruence

SAS (Side - Angle - Side) congruence criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

Step2: Analyze Given Information

We are given that $\angle DAC=\angle BCA$ and $AD = CB$. Let's consider triangles $\triangle DAC$ and $\triangle BCA$. The common side here is $AC$. For SAS, we need the sides adjacent to the given angle to be equal. In $\triangle DAC$, the sides adjacent to $\angle DAC$ are $AD$ and $AC$. In $\triangle BCA$, the sides adjacent to $\angle BCA$ are $CB$ and $AC$. We already know $AD = CB$ and $\angle DAC=\angle BCA$. The side $AC$ is common to both triangles (i.e., $AC = CA$ by the reflexive property of equality). Wait, but let's check the options. Wait, maybe I misread. Wait, the triangles are $\triangle DAC$ and $\triangle BCA$. Wait, $AD = CB$, $\angle DAC=\angle BCA$. The included angle for SAS: in $\triangle DAC$, the angle $\angle DAC$ is between $AD$ and $AC$. In $\triangle BCA$, the angle $\angle BCA$ is between $CB$ and $CA$. So we have $AD = CB$, $\angle DAC=\angle BCA$, and we need $AC = CA$ (which is always true as it's the same segment) but wait, no, maybe the triangles are $\triangle DAC$ and $\triangle BCA$. Wait, the options: let's check each option.

Option 1: $AB = CD$: This is a side of the parallelogram, but not related to the SAS for $\triangle DAC$ and $\triangle BCA$.

Option 2: $\angle CDA=\angle ACB$: This is an angle, not a side, so it doesn't help with SAS.

Option 3: $AC = CA$: This is the reflexive property, and it is the side that is included between the given side and angle. Since $AD = CB$, $\angle DAC=\angle BCA$, and $AC = CA$, by SAS, the triangles are congruent.

Option 4: $\angle B=\angle D$: This is an angle, not a side, so it doesn't help with SAS.

Wait, but actually, the common side $AC$ (which is $CA$) is the included side between the given side and angle. So the additional information needed (which is actually always true, but in the context of the options) is $AC = CA$.

Answer:

AC = CA (the option with "AC = CA")