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In the given figure, $ABCD$ is a rhombus with $∠ A = 67 °$ . If $Δ D E C$ is an equilateral triangle, then calculate $∠ CBE$ and $∠ DBE$ .

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∠ C B E = 36.5 °, ∠ D B E = 60 °

∠ C B E = 26.5 °, ∠ D B E = 30 °

∠ C B E = 26 °, ∠ D B E = 30.5 °

∠ C B E = 27 °, ∠ D B E = 20 °

answer is B.

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Detailed Solution

Given, a rhombus

ABCD

with

∠ A = 67 °

and an equilateral triangle

Δ D E C

In the rhombus ABCD,

∠ A = 67 ° = ∠ C

[Opposite angles of rhombus]

∠ A + ∠ D = 180 °

[Consecutive angles of rhombus.]

⇒ ∠ D = 113 ° ⇒ ∠ A B C = 113 ° [Opposite angles]

Δ D B C

D C = C B

[Sides of rhombus]

⇒ ∠ C D B = ∠ C B D

[Angles opposite to equal sides.]… (1)
Also, by angle sum property of triangle,

∠ C D B + ∠ C B D + ∠ B C D = 180 ° ⇒ 2 ∠ C D B + 67 ° = 180 ° [From (1)] ⇒ ∠ C D B = 56.5 ° ⇒ ∠ C D B = ∠ C B D = 56.5 ° ... (2)

Δ D C E

E C = C B

[Sides of equilateral triangle are equal.]

⇒ ∠ C B E = ∠ C E B

[Angles opposite to equal sides are equal]… (3)
By angle sum property of triangle,

∠ C B E + ∠ C E B + ∠ B C E = 180 ° ⇒ 2 ∠ C B E = 53 ° [From (2)] ⇒ ∠ C B E = 26.5 ° ... (4)

The value is

∠ C B E = 26.5 °

∠ C B D = ∠ C B E + ∠ D B E [Angle addition property .] ⇒ 56.5 = 26.5 + ∠ D B E [From (3) and (4)] ∴ ∠ D B E = 30 °

Hence,

∠ C B E = 26.5 °

and

∠ D B E = 30 °

.
Hence, the correct option is 2.

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