Question

in the given triangles, \\( \overline { d e } \cong \overline { x y } \\) and \\( \angle d \cong \angle x \\).
which additional fact is needed in order to use the asa criterion to prove that the two triangles are congruent?
\\( \bigcirc \\) a. \\( \overline { e f } \cong \overline { y z } \\)
\\( \bigcirc \\) b. \\( \angle f \cong \angle z \\)
\\( \bigcirc \\) c. \\( \overline { d f } \cong \overline { x z } \\)
\\( \bigcirc \\) d. \\( \angle e \cong \angle y \\)

Explanation:

Step1: Recall ASA criterion

ASA (Angle - Side - Angle) 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 two triangles are congruent.
We have \( \overline{DE}\cong\overline{XY}\) (a side) and \( \angle D\cong\angle X\) (an angle).

Step2: Analyze each option

• Option A: \( \overline{EF}\cong\overline{YZ}\) gives a non - included side, not relevant for ASA.
• Option B: \( \angle F\cong\angle Z\) gives a non - included angle, not relevant for ASA.
• Option C: \( \overline{DF}\cong\overline{XZ}\) gives a non - included side, not relevant for ASA.
• Option D: If \( \angle E\cong\angle Y\), then we have two angles (\( \angle D\cong\angle X\) and \( \angle E\cong\angle Y\)) and the included side \( \overline{DE}\cong\overline{XY}\) (since the side is between \( \angle D\) and \( \angle E\) in \(\triangle DEF\) and between \( \angle X\) and \( \angle Y\) in \(\triangle XYZ\)).

Answer:

D. \( \angle E\cong\angle Y\)