The figure shows triangle DEF and line segment BC, which is parallel to EF:
Triangle DEF has a point B on side DE and point C on side DF. The line BC is parallel to the line EF.
Part A: Is triangle DEF similar to triangle DBC? Explain using what you know about triangle similarity. (5 points)
Part B: Which line segment on triangle DBC corresponds to line segment EF? Explain your answer. (3 points)
Part C: Which angle on triangle DBC corresponds to angle F? Explain your answer. (2 points)
Part A: Yes, triangle DEF is similar to triangle DBC. Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. In this case, the angle DEF corresponds to angle DBC because they are both opposite the parallel lines EF and BC. Also, the angles DFE and BDC are congruent because they are corresponding angles formed by a transversal. Finally, the sides DE and DB, as well as EF and BC, are proportional because they are parallel lines intersected by transversals. Therefore, all the criteria for triangle similarity are met.
Part B: The line segment BC corresponds to the line segment EF. In similar triangles, corresponding sides are proportional. In this case, BC is parallel to EF and they are both opposite angles DBC and DEF. Therefore, BC is the corresponding side to EF.
Part C: Angle DBC corresponds to angle F. In similar triangles, corresponding angles are congruent. In this case, angle DBC is opposite side BC, which corresponds to side EF. Hence, angle DBC corresponds to angle F.