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

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

GP¯¯¯¯¯¯¯¯≅FP¯¯¯¯¯¯¯¯
line segment cap g cap p is congruent to line segment cap f cap p

∠FDP≅∠EPG
∠FDP≅∠EPG

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

∠DPF≅∠DPG

Answer 1

To deduce the correct statements from the information given, we need to analyze the properties of the perpendicular bisector and how it relates to the segments and angles involved.

Since ED is a perpendicular bisector of GF, it means:

1. **DP ≅ EP**: Since P is the midpoint of segment GF, segments DP and EP are congruent because a perpendicular bisector creates two equal segments from the midpoint.

2. **GP ≅ FP**: This statement is also true because the definition of a perpendicular bisector indicates that it divides the segment into two equal halves. Thus, GP and FP are congruent segments.

Now let's examine the possible options:

1. **DP ≅ EP** - This statement is true because P is the midpoint of GF.

2. **GP ≅ FP** - This statement is also true because P is the midpoint of GF.

3. **∠FDP ≅ ∠EPG** - This is not necessarily true since there is no direct implication about the angles formed by the perpendicular bisector and the segments.

4. **GF ≅ ED** - This is not true. While they are both segments, their lengths are not implied to be equal.

5. **∠DPF ≅ ∠DPG** - This is true as both angles are right angles (since ED is a perpendicular bisector), but it does not relate directly to the segments we are analyzing.

The definitive true statements based on the relationship between the perpendicular bisector and the segments are:

- **DP ≅ EP**
- **GP ≅ FP**

So, the correct selections are:

- DP ≅ EP
- GP ≅ FP