Characteristics and Proofs of a Rectangle: Identifying and defining elements

In this question, we are asked to find the measure of the angle marked in orange.

We can see that it is adjacent to the left vertex of the rectangle, and its sides are extensions of the rectangle’s sides. This means that the angle is directly opposite the rectangle’s vertex, where a right angle is formed.

Since all angles in a rectangle measure 90∘90∘, the angle we were asked to find is also equal to 90∘90∘.

One of the properties of a rectangle is that all its angles are right angles.

Therefore, it is not possible for an angle to be acute, that is, less than 90 degrees.

According to the side-angle-side theorem, the triangles are similar and coincide with each other:

AC = BD (Any pair of opposite sides of a parallelogram are equal)

Angle C is equal to angle B.

AB = CD (Any pair of opposite sides of the parallelogram are equal)

Therefore, all of the answers are correct.

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.

From the given information, we know that triangle EBC is equilateral.

In an equilateral triangle, all angles are equal to each other.

Therefore, angle B is equal to 60 degrees.

Since none of the angles are 90 degrees, we can safely say that the quadrilateral is not a rectangle.