Question

which steps will verify that a parallelogram is a rectangle? check all that apply.
calculate the lengths of the diagonals, and show that they are equal.
calculate the lengths of all sides, and show that both pairs of opposite sides are congruent.
calculate the slope of each diagonal, and show that the lines are perpendicular.
calculate the midpoints of each diagonal, and show the diagonals bisect each other.
calculate the slopes of every side, and show that adjacent sides are perpendicular.

Explanation:

1. For the option "Calculate the lengths of the diagonals, and show that they are equal": In a parallelogram, if the diagonals are equal in length, then the parallelogram is a rectangle (this is a theorem in geometry: A parallelogram with equal diagonals is a rectangle). So this step works.
2. For the option "Calculate the lengths of all sides, and show that both pairs of opposite sides are congruent": This is a property of all parallelograms (opposite sides of a parallelogram are congruent by definition), so this step just confirms it's a parallelogram, not necessarily a rectangle. So this is incorrect.
3. For the option "Calculate the slope of each diagonal, and show that the lines are perpendicular": If the diagonals are perpendicular, the parallelogram is a rhombus, not necessarily a rectangle. So this is incorrect.
4. For the option "Calculate the midpoints of each diagonal, and show the diagonals bisect each other": This is a property of all parallelograms (diagonals of a parallelogram bisect each other by definition), so this just confirms it's a parallelogram, not a rectangle. So this is incorrect.
5. For the option "Calculate the slopes of every side, and show that adjacent sides are perpendicular": If adjacent sides of a parallelogram are perpendicular, then all angles are right angles (since in a parallelogram, adjacent angles are supplementary, and if they are perpendicular, each is 90 degrees), so the parallelogram is a rectangle. So this step works.

Answer:

• Calculate the lengths of the diagonals, and show that they are equal.
• Calculate the slopes of every side, and show that adjacent sides are perpendicular.