Question

prove: $\triangle abc \cong \triangle def$

it is given that $\triangle abc$ and $\triangle def$ are both right triangles and have exactly one right angle. all right angles are congruent, so dropdown.

it is given that $\overline{ac} \cong \overline{df}$ and that $\overline{bc} \cong \overline{ef}$.

dropdown angle between $\overline{ac}$ and $\overline{bc}$.

dropdown angle between $\overline{df}$ and $\overline{ef}$.

through the congruencies of dropdown, $\overline{ac} \cong \overline{df}$, and $\overline{bc} \cong \overline{ef}$, it follows that $\triangle abc \cong \triangle def$ by the dropdown triangle congruency theorem.

Explanation:

Step1: Identify right angles congruence

In right triangles \( \triangle ABC \) and \( \triangle DEF \), the right angles are \( \angle C \) and \( \angle F \) respectively. Since all right angles are congruent, \( \angle C \cong \angle F \).

Step2: Identify included angles

For \( \triangle ABC \), the angle between \( \overline{AC} \) and \( \overline{BC} \) is \( \angle C \) (the right angle). For \( \triangle DEF \), the angle between \( \overline{DF} \) and \( \overline{EF} \) is \( \angle F \) (the right angle).

Step3: Apply SAS congruence

We have \( \overline{AC} \cong \overline{DF} \), \( \angle C \cong \angle F \), and \( \overline{BC} \cong \overline{EF} \). By the Side - Angle - Side (SAS) Triangle Congruency Theorem, \( \triangle ABC \cong \triangle DEF \).

Filling in the blanks:

1. The first blank: \( \angle C \cong \angle F \)
2. The second blank (angle between \( \overline{AC} \) and \( \overline{BC} \)): \( \angle C \) is the
3. The third blank (angle between \( \overline{DF} \) and \( \overline{EF} \)): \( \angle F \) is the
4. The fourth blank (congruent angle): \( \angle C \cong \angle F \)
5. The fifth blank (congruence theorem): SAS

Answer:

1. \( \angle C \cong \angle F \)
2. \( \angle C \)
3. \( \angle F \)
4. \( \angle C \cong \angle F \)
5. SAS