We are given that the diagonals of a square are along the lines y=x and y=- x+6. Let's put these lines on a coordinate plane and count squares to find the point of intersection.
Let's find points on the given lines that are equidistant from this intersection point.
Let's see why this construction guarantees that the resulting quadrilateral is a square.
The points are equidistant from the intersection point, so the diagonals bisect each other. According to Theorem 6.11, the quadrilateral is a parallelogram.
The points are equidistant from the intersection point, so the diagonals are congruent. According to Theorem 6.14, the parallelogram is a rectangle.
The given lines are perpendicular, so according to Theorem 6.17, the parallelogram is a rhombus.
Since the constructed quadrilateral is a rectangle and a rhombus, according to Theorem 6.20, it is a square.
