Question

it is given that \\( \angle brm \cong \angle thy \\), \\( \overline{mb} \cong \overline{yt} \\), and \\( \angle yth \cong \angle mbr \\).
the triangles \\( \text{select answer} \\) be proven congruent by \\( \text{select answer} \\).

Explanation:

Step1: Recall the ASA (Angle - Side - Angle) congruence criterion

The ASA 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 two triangles are congruent.

Step2: Identify the given congruent parts

We are given that \(\angle BRM\cong\angle THY\), \(\overline{MB}\cong\overline{YT}\), and \(\angle YTH\cong\angle MBR\).

Step3: Apply the ASA criterion

In \(\triangle MBR\) and \(\triangle YTH\), we have two angles (\(\angle MBR\) and \(\angle BRM\)) and the included side (\(\overline{MB}\)) in \(\triangle MBR\) congruent to two angles (\(\angle YTH\) and \(\angle THY\)) and the included side (\(\overline{YT}\)) in \(\triangle YTH\).

Answer:

The triangles \(\triangle MBR\) and \(\triangle YTH\) can be proven congruent by the ASA (Angle - Side - Angle) criterion.