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Answer :
Final answer:
Statements a, b, and c are true about squares: the diagonals are perpendicular, congruent to each other, and bisect the vertex angles. However, diagonals are not congruent to the sides nor twice the length of a side.
Explanation:
In reference to the question, the true statements about squares include the following: a. The diagonals are perpendicular. This fact can be proven using the properties of right triangles and orthogonal vectors. In a square, the diagonals intersect at a right angle, meaning they are perpendicular to each other.
b. The diagonals are congruent to each other. Given a square has all sides equal, and the Pythagorean theorem proves that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can conclude that the diagonals of a square are also equal, therefore congruent to each other.
c. The diagonals bisect the vertex angles. In a square, the diagonals intersect each other equally, dividing the square into four right triangles. Each vertex angle of the square is thus halved, making the diagonals bisectors of the vertex angles.
However, statements d and e are incorrect. The diagonals in a square are not congruent to the sides of the square nor are they twice the length of one side of the square.
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