We are given a square WXYZ and the point of intersection of the diagonals T. We want to find the measure of ∠ WTZ. Let's start by recalling that in a square, the diagonals are congruent and perpendicular, and that they bisect each other.
Since the diagonals are perpendicular, ∠WTZ is a right angle. Therefore, m∠WTZ=90.
Extra
Squares
Let's consider why the diagonals of a square are congruent and perpendicular and why they bisect each other. We will start by comparing the definitions of a square, rectangle, and rhombus.
Type of Quadrilateral
Definition
Square
A square is a parallelogram with four congruent sides and and four right angles.
Rectangle
A rectangle is a parallelogram with four right angles.
Rhombus
A rhombus is a parallelogram with four congruent sides.
We can see that the definition of a square overlaps with the definitions of both a rhombus and a rectangle. The Venn diagram summarizes the relationships between the parallelograms.
Since a square is a rectangle and a rhombus, all properties of rectangles and rhombuses apply to squares. The diagonals of a square are congruent and bisect each other because a square is a rectangle. The diagonals of a square are perpendicular because a square is a rhombus.
