Question

given that $\triangle abc$ and $\triangle def$ are both right triangles and have exactly one right angle. all right angles are congruent, so $\angle c\cong\angle f$.
is given that $\angle b\cong\angle e$ and that $\overline{ab}\cong\overline{de}$.
$ab$ is an included side between $\angle b$ and $\angle c$.
$de$ is an included side between $\angle e$ and $\angle f$.
through the congruencies of $\angle c\cong\angle f$, $\angle b\cong\angle e$, and $\overline{ab}\cong\overline{de}$, it follows that $\triangle abc\cong\triangle def$ by the triangle congruency theorem.
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Explanation:

Step1: Recall triangle congruence theorems

• Angle - Angle - Side (AAS): Two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle.
• Side - Side - Side (SSS): All three sides of one triangle are congruent to the three sides of another triangle.
• Side - Angle - Side (SAS): Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
• Angle - Side - Angle (ASA): Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.

Step2: Analyze the given congruences

We are given that \(\angle C\cong\angle F\), \(\angle B\cong\angle E\), and \(\overline{AB}\cong\overline{DE}\). The side \(\overline{AB}\) is between \(\angle B\) and \(\angle A\) (but we know \(\angle C\) is a right - angle. In \(\triangle ABC\), \(\angle A+\angle B+\angle C = 180^{\circ}\), and in \(\triangle DEF\), \(\angle D+\angle E+\angle F=180^{\circ}\)). Since \(\angle C\cong\angle F\) and \(\angle B\cong\angle E\), and \(\overline{AB}\) is a side. The side \(\overline{AB}\) is between \(\angle B\) and (the non - right angle \(\angle A\)), but when considering the congruence of \(\triangle ABC\) and \(\triangle DEF\) with \(\angle B\cong\angle E\), \(\angle C\cong\angle F\) (two angles) and \(\overline{AB}\cong\overline{DE}\) (the side is between the two angles \(\angle B\) and \(\angle C\) in \(\triangle ABC\) and between \(\angle E\) and \(\angle F\) in \(\triangle DEF\)).

Answer:

Angle - Side - Angle