Question

consider δwxy and δbcd with ∠x ≅∠c, \\(overline{wx} \cong \overline{bc}\\), and \\(overline{wy} \cong \overline{bd}\\). can it be concluded that \\( \triangle wxy \cong \triangle bcd \\) by sas? why or why not? \\(\bigcirc\\) no, because the third corresponding sides must also be given as congruent \\(\bigcirc\\) no, because the corresponding congruent angles listed are not the included angles \\(\bigcirc\\) no, because all corresponding angles must be given as congruent \\(\bigcirc\\) yes, because two corresponding sides and a corresponding angle are congruent

Explanation:

To determine if \(\triangle WXY \cong \triangle BCD\) by SAS, we recall the SAS (Side - Angle - Side) congruence criterion. The SAS 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.

In \(\triangle WXY\), the sides adjacent to \(\angle X\) are \(\overline{WX}\) and \(\overline{XY}\). In \(\triangle BCD\), the sides adjacent to \(\angle C\) are \(\overline{BC}\) and \(\overline{CD}\). We are given that \(\angle X\cong\angle C\), \(\overline{WX}\cong\overline{BC}\), and \(\overline{WY}\cong\overline{BD}\). The angle \(\angle X\) is not the included angle between \(\overline{WX}\) and \(\overline{WY}\) (for \(\triangle WXY\)) and \(\angle C\) is not the included angle between \(\overline{BC}\) and \(\overline{BD}\) (for \(\triangle BCD\)). So, the congruent angle is not the included angle between the two congruent sides, and thus SAS cannot be applied.

Let's analyze the other options:

• Option 1: The SAS criterion does not require the third side to be congruent. So this is incorrect.
• Option 3: SAS does not require all corresponding angles to be congruent. So this is incorrect.
• Option 4: Just having two sides and a non - included angle congruent does not satisfy SAS. So this is incorrect.

Answer:

no, because the corresponding congruent angles listed are not the included angles