Parallelogram Analysis: Determining if a 45° Quadrilateral is a Rectangle

According to the data shown in the drawing, we know that $AD=AB$

$BC=CD$

This means that triangles BCD and DAB are isosceles triangles.

Since we are given that angle ABD equals 45 degrees, angle ADB also equals 45 degrees (in an isosceles triangle, the base angles are equal)

We can calculate angle A since in a triangle the sum of angles equals 180:

$A+45+45=180$

$A+90=180$

$A=180-90$

$A=90$

Given that angle A is a right angle in the parallelogram, we can determine that the parallelogram is a rectangle according to the rule:

A parallelogram with at least one right angle is a rectangle.