Question

solve and choose the two correct answers carefully on why¯¯¯¯¯¯ 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 ∠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

When a line segment is the perpendicular bisector of another line segment, it has certain properties. In this case, if line segment \(ED\) is a perpendicular bisector of line segment \(GF\) and they intersect at point \(P\), the following statements must be true:

1. **Angle congruence**: Since \(ED\) is a perpendicular bisector, it means it divides \(GF\) into two equal lengths and intersects at a right angle. Thus, we can conclude that:
\[
\angle DPF \cong \angle DPG
\]

2. **Segment congruence**: The definition of a bisector states that the point of intersection divides the segment into two equal parts:
\[
GP \cong PF
\]

Thus, the two correct statements from the provided options are:
1. \( \angle DPF \cong \angle DPG \)
2. \( GP \cong PF \)

These reflect the properties of perpendicular bisectors and the congruence of segments created by the bisector.