Question

ED¯¯¯¯¯¯¯¯
is a perpendicular bisector of GF¯¯¯¯¯¯¯¯
, and they intersect at point P. Which of the following statements must be true? Select the TWO answers that are correct.(1 point)
Responses which two answers are correct and why

∠DPF≅∠DPG
∠DPF≅∠DPG

DP¯¯¯¯¯¯¯¯≅EP¯¯¯¯¯¯¯¯
line segment cap d cap p is congruent to line segment cap e cap p

GF¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯
line segment cap g cap f is congruent to line segment cap e cap d

∠FDP≅∠EPG
∠FDP≅∠EPG

GP¯¯¯¯¯¯¯¯≅FP¯¯¯¯¯¯¯¯

Answer 1

Given that \( \overline{ED} \) is a perpendicular bisector of \( \overline{GF} \), we can infer several properties about the relationships between the segments and angles involved in the scenario. A perpendicular bisector has two key features:

1. It bisects the segment into two equal parts.
2. It forms right angles with the segment it bisects.

Based on this, let's evaluate each option:

1. **\( \angle DPF \cong \angle DPG \)**: This statement is **true** because \( \overline{ED} \) being the perpendicular bisector means that it creates two right angles. Therefore, \( \angle DPF \) and \( \angle DPG \) are both right angles and therefore congruent.

2. **\( \overline{DP} \cong \overline{EP} \)**: This statement is **false** because while \( \overline{ED} \) is a perpendicular bisector, it indicates that the two line segments \( \overline{GF} \) are equal (not \( \overline{DP} \) and \( \overline{EP} \)).

3. **\( \overline{GF} \cong \overline{ED} \)**: This statement is **false** because \( \overline{ED} \) is the perpendicular bisector, but that does not imply that \( \overline{GF} \) is congruent to \( \overline{ED} \).

4. **\( \angle FDP \cong \angle EPG \)**: This statement is **false**. Although both angles share a common vertex at \( P \) and are formed by line \( \overline{ED} \), they are not congruent to each other. One is an angle formed off of \( \overline{GF} \) and the other off of \( \overline{GF} \) as well, but they are not equal as they correspond to different segments.

5. **\( \overline{GP} \cong \overline{FP} \)**: This statement is **true** because \( \overline{ED} \) being the perpendicular bisector means point \( P \) is the midpoint of \( \overline{GF} \), thus splitting \( \overline{GF} \) into two equal parts.

### Conclusion:
The two correct statements are:
- \( \angle DPF \cong \angle DPG \)
- \( \overline{GP} \cong \overline{FP} \)

These relationships stem from the properties of a perpendicular bisector in Euclidean geometry.