Question

EFGH is a parallelogram and angle HEF is a right angle. Select all statements that must be true.

The options are



1. EFGH is a rectangle



2. Triangle HEF is congruent to triangle GFH



3. Triangle HEF is congruent to triangle FGH



4. ED is congruent to HD, DG, and DF



5. Triangle EDH is congruent to triangle HDG



Please choose what are correct about the parallelogram :)

Answer 1

In a parallelogram with one right angle, EFGH must indeed be a rectangle, and triangle HEF is congruent to triangle GFH due to the hypotenuse-leg congruence theorem. Other statements either do not necessarily hold true or are unrelated to the given parallelogram.

Step-by-step explanation:

Given that EFGH is a parallelogram with a right angle at HEF, we can deduce certain properties about it:

• EFGH is a rectangle: If one angle of a parallelogram is a right angle, all four angles must be right angles, making it a rectangle. Thus, statement 1 is true.
• Triangle HEF is congruent to triangle GFH: Since EFGH is a rectangle, sides EH and FG are congruent, and HEF is a right angle, triangles HEF and GFH are right triangles sharing a common hypotenuse EF and one congruent side. Therefore, triangle HEF is congruent to triangle GFH by the hypotenuse-leg congruence theorem. This makes statement 2 true.
• Triangle HEF is congruent to triangle FGH is not necessarily true because statement 3 assumes that sides HE and FG are congruent, but without additional information that they are equal in length, we cannot conclude congruence.
• The symbols labelled ED, HD, DG, and DF do not correspond to any known sides of the parallelogram EFGH in standard notation, and this statement appears to be unrelated to the given question. Therefore, statement 4 is disregarded.
• Statement 5 cannot be true without additional information because we do not have sides or angles labelled as ED, HD, or DG in the parallelogram EFGH, nor do we have more context to assume congruence between triangles EDH and HDG.

Answer 2

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.