Question

In the adjoining figure, a rectangle ABCD is shown. If CAB = 20°, then find the values of x and y.

Answer 1

Given: Rectangle ABCD with AB ⟂ AD, AB || CD, AD || BC and ∠CAB = 20° (call θ = 20°).

Step-wise calculation:

1. AC makes an angle θ = 20° with AB.   ...(Given) 

Thus, AC makes 90° – θ = 90° – 20° = 70° with AD because AB ⟂ AD.

2. By symmetry or by placing A at the origin with AB along the x-axis and AD along the y-axis, the diagonal BD is directed at 180° – θ from the x-axis.

So, the angle between BD and the vertical side BC, which is at 90° is 

|(180° – θ) – 90°| 

= 90° – θ

= 70° 

Hence, x = 70°.

3. The two diagonals have directions θ for AC and 180° – θ for BD; the angle between them is

(180° – θ) – θ

= 180° – 2θ 

= 140°

So, the acute angle at their intersection is the supplement:

180° – 140° = 40°

Equivalently, that acute angle = 2θ = 40°.

Hence, y = 40°.