Question

dh 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)
∠dhf ≅ ∠dhg
fg ≅ de
ph ≅ gh
∠fdh ≅ ∠dgh
dh ≅ eh

Explanation:

Step1: Recall the property of perpendicular bisector

If a line is a perpendicular bisector of a segment, then any point on the perpendicular bisector is equidistant from the endpoints of the segment. But here, since \(DH\) is a perpendicular bisector of \(FG\), and we are dealing with angles and segments related to the intersection at \(H\).
When two lines are perpendicular, the adjacent angles formed are congruent. So, \(\angle DHF\) and \(\angle DHG\) are right - angles (because \(DH\) is a perpendicular bisector of \(FG\)), and \(\angle DHF\cong\angle DHG\) (by the definition of perpendicular lines: if \(a\perp b\), then the angles formed are \(90^{\circ}\) and all \(90^{\circ}\) angles are congruent).

Step2: Analyze the other options

• For \(\overline{FG}\cong\overline{DE}\), there is no information given in the problem about the relationship between \(FG\) and \(DE\) to support this congruence.
• For \(\overline{PH}\cong\overline{GH}\), there is no mention of point \(P\) in the problem description related to the property of the perpendicular bisector of \(FG\) (the problem only mentions \(DH\) as the perpendicular bisector of \(FG\)).
• For \(\angle FDH\cong\angle DGH\), there is no geometric theorem (such as triangle congruence theorems like ASA, SAS, SSS) or property (like angle - angle relationships from parallel lines etc.) given the information of \(DH\) being a perpendicular bisector of \(FG\) to support this angle - angle congruence.
• For \(\overline{DH}\cong\overline{EH}\), there is no information about point \(E\) in relation to the perpendicular bisector of \(FG\) to justify this segment congruence.

Answer:

\(\angle DHF\cong\angle DHG\) and \(\overline{FG}\cong\overline{DE}\) (assuming there was a mis - label and the intended segment for congruence based on perpendicular bisector properties was not mis - written. If we strictly go by the given options and the property of perpendicular lines (since \(DH\perp FG\) as it is a perpendicular bisector), the most certain one is \(\angle DHF\cong\angle DHG\))