Question

determining if triangles are congruent
consider △wxy and △bcd with ∠x ≅ ∠c, (overline{wx} cong overline{bc}), and (overline{wy} cong overline{bd}).
can it be concluded that △wxy ≅ △bcd by sas? why or why not?

• yes, because two corresponding sides and a corresponding angle are congruent
• no, because the third corresponding sides must also be given as congruent
• no, because the corresponding congruent angles listed are not the included angles
• no, because all corresponding angles must be given as congruent

Explanation:

Step1: Recall SAS Congruence Criterion

The SAS (Side - Angle - Side) congruence 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. The included angle is the angle between the two given sides.

Step2: Analyze the Given Information

In \(\triangle WXY\) and \(\triangle BCD\), we have \(\angle X\cong\angle C\), \(\overline{WX}\cong\overline{BC}\), and \(\overline{WY}\cong\overline{BD}\). Let's check the sides and the angle:

• For \(\triangle WXY\), the sides adjacent to \(\angle X\) are \(\overline{WX}\) and \(\overline{XY}\).
• For \(\triangle BCD\), the sides adjacent to \(\angle C\) are \(\overline{BC}\) and \(\overline{CD}\).
• We are given \(\overline{WX}\cong\overline{BC}\) and \(\overline{WY}\cong\overline{BD}\), but \(\angle X\) is not the included angle between \(\overline{WX}\) and \(\overline{WY}\) (the included angle between \(\overline{WX}\) and \(\overline{WY}\) would be \(\angle W\), not \(\angle X\)), and \(\angle C\) is not the included angle between \(\overline{BC}\) and \(\overline{BD}\) (the included angle between \(\overline{BC}\) and \(\overline{BD}\) would be \(\angle B\), not \(\angle C\)). So the congruent angle is not the included angle between the two congruent sides.

So, we cannot conclude that \(\triangle WXY\cong\triangle BCD\) by SAS because the corresponding congruent angles listed are not the included angles.

Answer:

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