Question

which statements are true of all squares? choose three correct answers
the diagonals are perpendicular.
the diagonals are congruent to the sides of the square.
the diagonals bisect the vertex angles.
the diagonals are twice the length of one side of the square.
the diagonals are congruent to each other.

Explanation:

• Diagonals are perpendicular: In a square, the diagonals intersect at \(90^{\circ}\), so they are perpendicular.
• Diagonals bisect the vertex angles: Each diagonal of a square divides a \(90^{\circ}\) vertex angle into two \(45^{\circ}\) angles.
• Diagonals are congruent to each other: Using the Pythagorean theorem \(d = \sqrt{s^{2}+s^{2}}=\sqrt{2}s\) (where \(s\) is the side - length of the square), both diagonals have the same length.

The statement "The diagonals are congruent to the sides of the square" is false because if the side - length is \(s\), the diagonal \(d=\sqrt{2}s
eq s\). The statement "The diagonals are twice the length of one side of the square" is false since \(d = \sqrt{2}s
eq2s\).

Answer:

• The diagonals are perpendicular.
• The diagonals bisect the vertex angles.
• The diagonals are congruent to each other.