¯DE¯ is a perpendicular bisector of ¯FG¯ , and they intersect at point H. Which of the following statements must be true? Select the two correct answers.
¯DH¯≅¯EH¯
¯FG¯≅¯DE¯
∠DHF≅∠DHG
∠FDH≅∠DGH
¯FH¯≅¯GH¯
The correct statements are:
1) ¯DH¯≅¯EH¯: Since ¯DE¯ is the perpendicular bisector of ¯FG¯, point H is the midpoint of ¯FG¯. Therefore, ¯DH¯ and ¯EH¯, which are segments from H to the endpoints of ¯FG¯, will be congruent.
2) ¯FG¯≅¯DE¯: Since ¯DE¯ is the perpendicular bisector of ¯FG¯, it divides ¯FG¯ into two congruent segments. Therefore, ¯FG¯ and ¯DE¯ will be congruent.
The other statements are not necessarily true:
3) ∠DHF≅∠DHG: We don't have enough information to determine the congruence of these angles. The fact that ¯DE¯ is the perpendicular bisector of ¯FG¯ only tells us about the lengths of the segments, not the angles.
4) ∠FDH≅∠DGH: We don't have enough information to determine the congruence of these angles. The fact that ¯DE¯ is the perpendicular bisector of ¯FG¯ only tells us about the lengths of the segments, not the angles.
5) ¯FH¯≅¯GH¯: We don't have enough information to determine the lengths of these segments. The fact that ¯DE¯ is the perpendicular bisector of ¯FG¯ only tells us that H is the midpoint of ¯FG¯, but it does not tell us the lengths of ¯FH¯ and ¯GH¯.