Question

In the given figure ABCD is a rhombus with angle A 67degree if DEC is an equilateral triangle calaculate ANGLE CBE AND ANGLE CBE

Answer 1

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Answer 2

∠ CBE = 26.5°  and ∠ DBE =  30°

Step-by-step explanation:

Given:

In rhombus ABCD, ∠ A = 67°

∴ ∠ BAD = 67° and ∠ ABC = 180° - 67° = 113°  

∆ DEC is an equilateral triangle

∴ ∠ DCE = 60°

∠ BCE = ∠ BCD + ∠ DCE = 67° + 60° = 127°            [1]

In ∆BCE,

BC = CE                   [ sides of equilateral triangle are equal]

∴ ∠ CBE = ∠ CEB   [ angles adjacent to equal sides are equal    [2]

Using angle sum property for ∆BCE  

∠ CBE + ∠ CEB + ∠ BCE = 180°                

⇒ ∠CBE + ∠CBE + 127° = 180°  [using (1) and (2)]

⇒ 2∠ CBE = 180° - 127° = 53°

⇒ $\angle CBE = \frac{53 }{2}$

⇒ ∠ CBE = 26.5°

 

Diagonals of a rhombus bisect the angle ,

⇒ ∠ABD = ∠DBC = $\frac{\angle ABC}{2}$ = $\frac{113}{2}$ = 56.5°                

⇒ ∠ DBE = ∠ DBC - ∠ CBE = 56.5° – 26.5° = 30°

⇒ ∠ DBE =  56.5° – 26.5°

⇒ ∠ DBE =  30°

Therefore, ∠ CBE = 26.5°  and ∠ DBE =  30°