In the given figure, ABCD is rhombus and △EDC is a equilateral. If ∠BAD = 78°, calculate :
(i) ∠CBE
(ii) ∠DBE

Question

In the given figure, ABCD is rhombus and △EDC is a equilateral. If ∠BAD = 78°, calculate : (i) ∠CBE (ii) ∠DBE

KnowledgeBoat · Accepted Answer

(i) In rhombus ABCD, ⇒ ∠C = ∠A = 78° (Opposite angles of rhombus are equal) The sum of adjacent angles of a rhombus is always 180°. ⇒ ∠A + ∠B = 180° ⇒ 78° + ∠B = 180° ⇒ ∠B = 180° - 78° = 102°. In △ DBC, ⇒ DC = CB (Sides of rhombus are equal in length) .....(1) ⇒ ∠CDB = ∠CBD = x (let) (In a triangle angles opposite to equal sides are equal.) By angle sum property of triangle, ⇒ ∠CDB + ∠CBD + ∠BCD = 180° ⇒ x + x + ∠BCD = 180° ⇒ 2x + 78° = 180° ⇒ 2x = 180° - 78° ⇒ 2x = 102° ⇒ x = ⇒ ∠CDB = ∠CBD = 51°. Given, DEC is an equilateral triangle, so all the sides of triangle are equal. ∴ DC = EC .......(2) From equations (1) and (2), we get : ⇒ CB = EC In triangle CEB, ⇒ ∠CEB = ∠CBE = y (let) [Angles opposite to equal sides in a triangle are equal] By angle sum property of triangle, ⇒ ∠CEB + ∠CBE + ∠ECB = 180° ⇒ y + y + (∠ECD + ∠BCD) = 180° ⇒ 2y + (60° + 78°) = 180° ⇒ 2y + 138° = 180° ⇒ 2y = 180° - 138° ⇒ 2y = 42° ⇒ y = ⇒ ∠CBE = 21°. Hence, ∠CBE = 21°. (ii) From figure, ⇒ ∠DBE = ∠CBD - ∠CBE ⇒ ∠DBE = 51° - 21° = 30°. Hence, ∠DBE = 30°.

Mathematics

In the given figure, ABCD is rhombus and △EDC is a equilateral. If ∠BAD = 78°, calculate :
(i) ∠CBE
(ii) ∠DBE

Rectilinear Figures

Related Questions

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Mathematics

In the given figure, ABCD is rhombus and △EDC is a equilateral. If ∠BAD = 78°, calculate :(i) ∠CBE(ii) ∠DBE

Rectilinear Figures

Related Questions

In the given figure, ABCD is an isosceles trapezium in which ∠CDA = 2x° and ∠BAD = 3x°. Find all the angles of the trapezium.

In the given figure, ABCD is a trapezium in which ∠A = (x + 25)°, ∠B = y°, ∠C = 95° and ∠D = (2x + 5)°. Find the values of x and y.

DEC is an equilateral triangle in a square ABCD. If BD and CE intersect at O and ∠COD = x°, find the value of x.

If one angle of a parallelogram is 90°, show that each of its angles measures 90°.

In the given figure, ABCD is rhombus and △EDC is a equilateral. If ∠BAD = 78°, calculate :
(i) ∠CBE
(ii) ∠DBE