Question

it is given that \\( \triangle abc \\) and \\( \triangle def \\) are both right triangle right angle. all right angles are congruent, so \\( \angle c \cong \angle f \\) it is given that \\( \angle b \cong \angle e \\) and that \\( \overline{ab} \cong \overline{de} \\). \\( \overline{ab} \\) is side between \\( \angle b \\) and \\( \angle c \\). \\( \overline{de} \\) is side between \\( \angle e \\) and \\( \angle f \\). through follows theore...

Explanation:

Step1: Analyze the relationship of sides and angles

In triangle congruence, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the two triangles are congruent by the Angle - Side - Angle (ASA) criterion. Here, \(\overline{AB}\) is the included side between \(\angle B\) and \(\angle C\) in \(\triangle ABC\), and \(\overline{DE}\) is the included side between \(\angle E\) and \(\angle F\) in \(\triangle DEF\).

Step2: Determine the congruence criterion

Since \(\angle B\cong\angle E\), \(\angle C\cong\angle F\) (right angles), and \(\overline{AB}\cong\overline{DE}\), by the Angle - Side - Angle (ASA) congruence theorem, \(\triangle ABC\cong\triangle DEF\).

Answer:

The first blank should be filled with “an included”, and the second blank should be filled with “ASA (Angle - Side - Angle)”.