Answer:
below :)
Explanation:
o prove the statement "If a quadrilateral is a square, then the quadrilateral is a rectangle," we need to show that any square must also be a rectangle. In geometry, a rectangle is a quadrilateral with all angles equal to 90 degrees. A square is a specific type of rectangle in which all sides are equal in length.
Proof:
Let's assume we have a quadrilateral that is a square. We'll denote this square as ABCD, where A, B, C, and D are the four vertices of the square.
Definition of a Square:
By definition, a square is a quadrilateral with all sides equal in length and all angles equal to 90 degrees.
Opposite Sides are Parallel:
In a square, opposite sides are parallel to each other. This property is common to all parallelograms, and a square is a special type of parallelogram.
Diagonals of a Square:
The diagonals of a square bisect each other at 90 degrees. This property is unique to a square.
Now, to prove that a square is a rectangle, we need to show that all angles of the square are 90 degrees.
Proof by Contradiction:
Assume that one angle of the square is not 90 degrees. Without loss of generality, let's assume that angle A is not 90 degrees.
In a square, all sides are equal. Therefore, AB = BC = CD = DA.
Now, consider the two triangles, ΔABC and ΔCDA.
Since AB = BC and CD = DA, we have two sides of the triangles equal.
AB = BC => Side-Angle-Side (SAS) congruence
CD = DA => Side-Angle-Side (SAS) congruence
Now, we have ΔABC ≅ ΔCDA by congruence.
According to the properties of congruent triangles, the corresponding angles of congruent triangles are equal. Therefore, angle C of ΔABC and angle A of ΔCDA are equal.
Since angle A is not 90 degrees (our assumption), angle C cannot be 90 degrees either. But this contradicts the fact that all angles of a square are 90 degrees.
Since our assumption leads to a contradiction, it must be false. Thus, the assumption that one angle of the square is not 90 degrees is incorrect.
Therefore, all angles of the square are 90 degrees, which means the square is a rectangle.
Hence, the statement is proved: "If a quadrilateral is a square, then the quadrilateral is a rectangle."