Question

in exercises 9-12, decide whether you can use the given information to prove that $\triangle abc cong \triangle def$. explain your reasoning.

1. $angle a cong angle d, angle c cong angle f, overline{ac} cong overline{df}$
2. $angle c cong angle f, overline{ab} cong overline{de}, overline{bc} cong overline{ef}$

Explanation:

For Exercise 9:

We are given two pairs of congruent angles ($\angle A \cong \angle D$, $\angle C \cong \angle F$) and a pair of congruent included sides ($\overline{AC} \cong \overline{DF}$). This matches the Angle-Side-Angle (ASA) triangle congruence criterion, which states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

For Exercise 10:

We are given one pair of congruent angles ($\angle C \cong \angle F$) and two pairs of congruent sides ($\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$). The congruent angle is not the included angle between the two congruent sides, so this does not fit a valid triangle congruence criterion (SSA is not a valid method to prove triangle congruence, as it can result in non-congruent triangles).

Answer:

1. Yes, $\triangle ABC \cong \triangle DEF$ by the ASA congruence criterion.
2. No, the given information does not satisfy a valid triangle congruence criterion (SSA is not a valid proof method).