Final answer:
In a parallelogram EFGH where angle HEF is a right angle, it can only be definitively concluded that EFGH is a rectangle. The congruence of triangles HEF and GFH is uncertain, and the statements concerning a point D cannot be verified without additional information.
Step-by-step explanation:
The question pertains to the properties of a parallelogram where one angle is given as a right angle. Given EFGH is a parallelogram and angle HEF is a right angle, let us evaluate the statements that must be true.
- EFGH is a rectangle - This statement is true. If one angle of a parallelogram is a right angle, then all the other angles must also be right angles, making the parallelogram a rectangle.
- Triangle HEF is congruent to triangle GFH - This statement could be true if we consider diagonals, but without knowing the specific measures or additional properties of the diagonals, we cannot conclusively say that these triangles are congruent. This statement is uncertain based on the given information.
- Triangle HEF is congruent to triangle FGH - This statement is false. Triangles HEF and FGH share a common side (FH), but there is no information provided that would lead to the conclusion they are congruent.
- ED is congruent to HD, DG, and DF - Without further context, it is unclear what point D refers to or how it relates to the parallelogram EFGH. This statement lacks sufficient information to be judged as true or false.
- Triangle EDH is congruent to triangle HDG - As with the previous statement, without more information regarding point D and its relationship with the parallelogram, we cannot determine the truth of this statement.
Overall, based on the properties of a parallelogram and the right angle present, we can only definitively conclude that EFGH is a rectangle.