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.(1 point)
Responses
DH¯¯¯¯¯¯¯¯¯≅EH¯¯¯¯¯¯¯¯
line segment cap d cap h is congruent to line segment cap e cap h
∠DHF≅∠DHG
∠DHF≅∠DHG
∠FDH≅∠DGH
∠FDH≅∠DGH
FH¯¯¯¯¯¯¯¯≅GH¯¯¯¯¯¯¯¯
line segment cap f cap h is congruent to line segment cap g cap h
FG¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯
FG¯¯¯¯¯¯¯¯ line segment cap f cap g is congruent to line segment cap d cap e
whats the other one
The other correct statement is:
∠DHF ≅ ∠DHG.
To determine which of the statements must be true, we need to analyze the given information and apply the properties of perpendicular bisectors. Here's how we can do that:
1. DE¯¯¯¯¯¯¯¯ is a perpendicular bisector of FG¯¯¯¯¯¯¯¯: This means that DE¯¯¯¯¯¯¯¯ intersects FG¯¯¯¯¯¯¯¯ at a right angle and also divides FG¯¯¯¯¯¯¯¯ into two equal parts.
From this information, we can conclude the following:
- DH¯¯¯¯¯¯¯¯¯≅EH¯¯¯¯¯¯¯¯: This is true since DE¯¯¯¯¯¯¯¯ is the perpendicular bisector of FG¯¯¯¯¯¯¯¯. The points D and E are equidistant from the midpoint H, which means that DH¯¯¯¯¯¯¯¯¯ is congruent to EH¯¯¯¯¯¯¯¯.
- ∠DHF≅∠DHG: This statement cannot be determined from the given information. Without additional information about the angles or specific properties of the figure, we cannot conclude that ∠DHF is congruent to ∠DHG.
- ∠FDH≅∠DGH: This statement cannot be determined from the given information. Again, we lack specific angle properties or additional information to conclude that ∠FDH is congruent to ∠DGH.
- FH¯¯¯¯¯¯¯¯≅GH¯¯¯¯¯¯¯¯: This is also true since DE¯¯¯¯¯¯¯¯ is the perpendicular bisector of FG¯¯¯¯¯¯¯¯. It divides FG¯¯¯¯¯¯¯¯ into two congruent parts, and FH¯¯¯¯¯¯¯¯ is equal to GH¯¯¯¯¯¯¯¯.
- FG¯¯¯¯¯¯¯¯≅DE¯¯¯¯¯¯¯¯: This statement is not necessarily true. Although DE¯¯¯¯¯¯¯¯ is a perpendicular bisector of FG¯¯¯¯¯¯¯¯, it only guarantees that DE¯¯¯¯¯¯¯¯ and FG¯¯¯¯¯¯¯¯ intersect at a right angle and divide FG¯¯¯¯¯¯¯¯ into two equal parts. It does not necessarily mean that FG¯¯¯¯¯¯¯¯ is congruent to DE¯¯¯¯¯¯¯¯.
Based on our analysis, the two correct statements are:
1. DH¯¯¯¯¯¯¯¯¯≅EH¯¯¯¯¯¯¯¯
2. FH¯¯¯¯¯¯¯¯≅GH¯¯¯¯¯¯¯¯