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 GP¯¯¯¯¯¯¯¯≅FP¯¯¯¯¯¯¯¯ line segment cap g cap p is congruent to line segment cap f cap p ∠DPF≅∠DPG ∠DPF≅∠DPG ∠FDP≅∠EPG ∠FDP≅∠EPG GF¯¯¯¯¯¯¯¯≅ED¯¯¯¯¯¯¯¯ line segment cap g cap f is congruent to line segment cap e cap d DP¯¯¯¯¯¯¯¯≅EP¯¯¯¯¯¯¯¯
Answers
Answered by
GPT-4o mini
Given that line segment \( ED \) is the perpendicular bisector of line segment \( GF \), we can derive some properties and relationships that must hold true.
1. **\( GP \cong FP \)**: This statement must be true due to the definition of a perpendicular bisector. Since \( ED \) is the perpendicular bisector of \( GF \), the point of intersection \( P \) divides \( GF \) into two equal segments.
2. **\( \angle DPF \cong \angle DPG \)**: This statement, however, does not hold true. The angles at point \( P \) are not necessarily equal only because \( ED \) is the perpendicular bisector.
3. **\( GF \cong ED \)**: This statement is also not true. The lengths of \( GF \) and \( ED \) are not necessarily equal just because \( ED \) is a perpendicular bisector.
4. **\( \angle FDP \cong \angle EPG \)**: This is true because \( ED \) being the perpendicular bisector implies that angles created by the segment with the bisector will be equal.
5. **\( DP \cong EP \)**: This statement is not necessarily true. The segments \( DP \) and \( EP \) are not equal simply because they lie on different segments defined by the perpendicular bisector.
Thus, the two statements that must be true are:
- \( GP \cong FP \)
- \( \angle FDP \cong \angle EPG \)
In conclusion, the responses that are correct are:
1. \( GP \cong FP \)
2. \( \angle FDP \cong \angle EPG \)
1. **\( GP \cong FP \)**: This statement must be true due to the definition of a perpendicular bisector. Since \( ED \) is the perpendicular bisector of \( GF \), the point of intersection \( P \) divides \( GF \) into two equal segments.
2. **\( \angle DPF \cong \angle DPG \)**: This statement, however, does not hold true. The angles at point \( P \) are not necessarily equal only because \( ED \) is the perpendicular bisector.
3. **\( GF \cong ED \)**: This statement is also not true. The lengths of \( GF \) and \( ED \) are not necessarily equal just because \( ED \) is a perpendicular bisector.
4. **\( \angle FDP \cong \angle EPG \)**: This is true because \( ED \) being the perpendicular bisector implies that angles created by the segment with the bisector will be equal.
5. **\( DP \cong EP \)**: This statement is not necessarily true. The segments \( DP \) and \( EP \) are not equal simply because they lie on different segments defined by the perpendicular bisector.
Thus, the two statements that must be true are:
- \( GP \cong FP \)
- \( \angle FDP \cong \angle EPG \)
In conclusion, the responses that are correct are:
1. \( GP \cong FP \)
2. \( \angle FDP \cong \angle EPG \)