Question

select all the correct answers. which statements are always true for a rectangle? all sides are congruent. the diagonals of a rectangle are perpendicular to each other. the diagonals are congruent. all angles are congruent. the opposite sides are parallel.

Explanation:

Step1: Analyze side - congruence

In a rectangle, only opposite sides are congruent, not all sides. So "All sides are congruent" is false.

Step2: Analyze diagonal - perpendicularity

The diagonals of a rectangle are not perpendicular to each other (except in the special case of a square, which is a special rectangle). So "The diagonals of a rectangle are perpendicular to each other" is false.

Step3: Analyze diagonal - congruence

By the properties of a rectangle, the diagonals are congruent. Let the length of the rectangle be \(l\) and width be \(w\). Using the Pythagorean theorem, if the diagonals are \(d_1\) and \(d_2\), \(d_1=\sqrt{l^{2}+w^{2}}\) and \(d_2=\sqrt{l^{2}+w^{2}}\), so \(d_1 = d_2\). The statement "The diagonals are congruent" is true.

Step4: Analyze angle - congruence

All angles in a rectangle are right - angles (\(90^{\circ}\)). So all angles are congruent. The statement "All angles are congruent" is true.

Step5: Analyze side - parallelism

By the definition of a rectangle (a parallelogram with four right - angles), opposite sides are parallel. The statement "The opposite sides are parallel" is true.

Answer:

The diagonals are congruent.
All angles are congruent.
The opposite sides are parallel.