Question

ABCD is a rectangle. If ABP and BCQ are equilateral triangles, find the $\angle $PBQ.

(a) 65°
(b) 75°
(c) 60°
(d) 90°

Answer 1

Hint: It is given in the question that ABCD is a rectangle and we know that all the angles of a rectangle are right angles. It is also known that triangle ABP and triangle BCQ are equilateral triangles. We know that each angle of an equilateral triangle is 60°. With this information, we will find the $\angle $PBC and add it to $\angle $CBQ to find $\angle $PBQ.

Complete step-by-step answer:
The figure for the question is as follows:

ABCD is a rectangle and each angle of a rectangle is 90°
$\angle $ABC = $\angle $BCD = $\angle $CDA = $\angle $DAB = 90°
Triangle ABP and triangle BCQ are equilateral triangle and each angle of an equilateral triangle is 60°.
$\angle $ABP = $\angle $BPA = $\angle $PAB = $\angle $BQC = $\angle $QCB = $\angle $CBQ = 60°
From the figure, we can deduce that
$\angle $ABC = $\angle $ABP + $\angle $PBC
$\Rightarrow $ 90° = 60° + $\angle $PBC
$\Rightarrow $ $\angle $PBC = 90° ─ 60°
$\Rightarrow $ $\angle $PBC = 30°
We know that, $\angle $CBQ = 60°.
And from the figure, we know that $\angle $PBQ = $\angle $PBC + $\angle $CBQ = 30° + 60° = 90°
Therefore, $\angle $PBQ = 90°

Note: Students are advised to be well-versed with the properties of various standard shapes. This question is relatively simple if the student is well acquainted with the concepts of geometry.