1. **Stating the problem:** We are given two triangles, \(\triangle DEF\) and \(\triangle ABC\), with \(\angle F \cong \angle C\) and right angles at \(D\) and \(A\) respectively. We want to determine which similarity statements are true based on this information. 2. **Recall the AA similarity postulate:** Two triangles are similar if two pairs of corresponding angles are congruent. 3. **Identify the angles:** - \(\angle D = 90^\circ\) and \(\angle A = 90^\circ\) (both right angles). - \(\angle F \cong \angle C\) (given). 4. **Apply AA similarity:** Since two pairs of angles are congruent, \(\triangle DEF \sim \triangle ABC\) by the AA similarity postulate. 5. **Check the order of vertices:** The correspondence of angles is \(D \leftrightarrow A\), \(E \leftrightarrow B\), and \(F \leftrightarrow C\). So the similarity is \(\triangle DEF \sim \triangle ABC\). 6. **Rewrite similarity to match answer options:** The order in the answer options is \(\triangle ABC\) first, so \(\triangle ABC \sim \triangle DEF\). 7. **Corresponding sides proportional:** Since the triangles are similar, corresponding sides are proportional: - \(AC \sim DF\) because \(C \leftrightarrow F\) - \(BC \sim DE\) because \(B \leftrightarrow E\) 8. **Summary of correct statements:** - \(\triangle ABC \sim \triangle DEF\) by AA similarity postulate - \(AC \sim DF\) because corresponding parts of similar triangles are proportional - \(BC \sim DE\) because corresponding parts of similar triangles are proportional 9. **Incorrect statement:** \(\triangle ABC \sim \triangle EDF\) is incorrect because the order of vertices does not match the angle correspondence. **Final answer:** - \(\triangle ABC \sim \triangle DEF\) because of the AA similarity postulate - \(AC \sim DF\) because corresponding parts of similar triangles are proportional - \(BC \sim DE\) because corresponding parts of similar triangles are proportional