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:
- It bisects the segment into two equal parts.
- It forms right angles with the segment it bisects.
Based on this, let's evaluate each option:
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\( \angle DPF \cong \angle DPG \): 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.
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\( \overline{DP} \cong \overline{EP} \): 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 \( \overline{EP} \)).
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\( \overline{GF} \cong \overline{ED} \): This statement is false because \( \overline{ED} \) is the perpendicular bisector, but that does not imply that \( \overline{GF} \) is congruent to \( \overline{ED} \).
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\( \angle FDP \cong \angle EPG \): 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.
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\( \overline{GP} \cong \overline{FP} \): 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.
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