Question

which statements are true of all squares? select three options.

the diagonals are perpendicular.
the diagonals are congruent to each other.
the diagonals bisect the vertex angles.
the diagonals are congruent to the sides of the square.
the diagonals are twice the length of one side of the square.

Explanation:

Analyze perpendicular diagonals

Using the Square Diagonal Properties knowledge point

$$ \text{Diagonals of a square intersect at right angles } (90^\circ). $$

Thus, the diagonals are perpendicular. This statement is true.

Analyze congruent diagonals

Using the Square Diagonal Properties knowledge point

$$ \text{Since a square is a rectangle, its diagonals are equal in length.} $$

Thus, the diagonals are congruent to each other. This statement is true.

Analyze vertex angle bisection

Using the Square Diagonal Properties knowledge point

$$ \text{Since a square is a rhombus, its diagonals bisect its } 90^\circ \text{ vertex angles into two } 45^\circ \text{ angles.} $$

Thus, the diagonals bisect the vertex angles. This statement is true.

Analyze diagonal to side length ratio

For a square with side length \(s\), we apply the Pythagorean theorem to find the diagonal length \(d\):

$$ d = \sqrt{s^2 + s^2} = s\sqrt{2} \approx 1.414s $$

Since \(s\sqrt{2}
eq s\) and \(s\sqrt{2}
eq 2s\), the diagonals are neither congruent to the sides nor twice the length of a side. Thus, the remaining two statements are false.

Answer:

• The diagonals are perpendicular. (Correct answer)
• The diagonals are congruent to each other. (Correct answer)
• The diagonals bisect the vertex angles. (Correct answer)
• The diagonals are congruent to the sides of the square.
• The diagonals are twice the length of one side of the square.