The given figure shows a square ABCD and an equilateral triangle ABP. Calculate :
(i) ∠AOB
(ii) ∠BPC
(iii) ∠PCD
(iv) reflex ∠APC

Question

The given figure shows a square ABCD and an equilateral triangle ABP. Calculate : (i) ∠AOB (ii) ∠BPC (iii) ∠PCD (iv) reflex ∠APC

KnowledgeBoat · Accepted Answer

(i) Given, ABP is an equilateral triangle. ∴ ∠PAB = 60° From figure, ⇒ ∠OAB = ∠PAB = 60°. We know that, Each interior angle of a square equals 90° and diagonals bisect interior angles. ∴ ∠DBA = = 45°. From figure, ⇒ ∠OBA = ∠DBA = 45°. In △ AOB, By angle sum property of triangle, ⇒ ∠OBA + ∠OAB + ∠AOB = 180° ⇒ 45° + 60° + ∠AOB = 180° ⇒ ∠AOB + 105° = 180° ⇒ ∠AOB = 180° - 105° = 75°. Hence, ∠AOB = 75°. (ii) From figure, ⇒ ∠PBA = 60° [Each angle of an equilateral triangle equals to 60°.] ⇒ ∠CBP = ∠CBA - ∠PBA = 90° - 60° = 30°. We know that, ⇒ BP = AB (Sides of equilateral triangle) .........(1) ⇒ AB = BC (Sides of square ABCD are equal) ........(2) From equation (1) and (2), we get : ⇒ BP = BC. In △ BPC, ⇒ BP = BC ⇒ ∠BCP = ∠BPC = x (let) [Angles opposite to equal sides are equal] By angle sum property of triangle, ⇒ ∠BCP + ∠BPC + ∠CBP = 180° ⇒ x + x + 30° = 180° ⇒ 2x = 180° - 30° ⇒ 2x = 150° ⇒ x = = 75°. Hence, ∠BPC = 75°. (iii) As, ⇒ ∠BCP = ∠BPC = 75° From figure, ⇒ ∠C = ∠BCP + ∠PCD ⇒ 90° = 75° + ∠PCD ⇒ ∠PCD = 90° - 75° = 15°. Hence, ∠PCD = 15°. (iv) From figure, ⇒ ∠APC = ∠APB + ∠BPC ⇒ ∠APC = 60° + 75° = 135° ⇒ Reflex ∠APC = 360° - ∠APC = 360° - 135° = 225°. Hence, reflex ∠APC = 225°.

Mathematics

The given figure shows a square ABCD and an equilateral triangle ABP. Calculate :
(i) ∠AOB
(ii) ∠BPC
(iii) ∠PCD
(iv) reflex ∠APC

Rectilinear Figures

Related Questions

In the following figure, AE and BC are equal and parallel and the three sides AB, CD and DE are equal to one another. If angle A is 102°. Find angles AEC and BCD.

In the given figure; ABCD is a rhombus with angle A = 67°. If DEC is an equilateral triangle, calculate :
(i) ∠CBE
(ii) ∠DBE

In each of the following figures, ABCD is a parallelogram.
(i)
(ii)
In each case, given above, find the values of x and y.

Mathematics

The given figure shows a square ABCD and an equilateral triangle ABP. Calculate :(i) ∠AOB(ii) ∠BPC(iii) ∠PCD(iv) reflex ∠APC

Rectilinear Figures

Related Questions

In the following figure, AE and BC are equal and parallel and the three sides AB, CD and DE are equal to one another. If angle A is 102°. Find angles AEC and BCD.

In the given figure; ABCD is a rhombus with angle A = 67°. If DEC is an equilateral triangle, calculate :(i) ∠CBE(ii) ∠DBE

In each of the following figures, ABCD is a parallelogram.(i)(ii)In each case, given above, find the values of x and y.

The given figure shows a square ABCD and an equilateral triangle ABP. Calculate :
(i) ∠AOB
(ii) ∠BPC
(iii) ∠PCD
(iv) reflex ∠APC

In the given figure; ABCD is a rhombus with angle A = 67°. If DEC is an equilateral triangle, calculate :
(i) ∠CBE
(ii) ∠DBE

In each of the following figures, ABCD is a parallelogram.
(i)
(ii)
In each case, given above, find the values of x and y.