Question

One of the diagonals of a Rhombus is equal to one of its sides . Find the angles of the Rhombus.​

Answer 1

$\large\underline{ \underline{ \sf \maltese{ \: Question:- }}}$

• One of the diagonals of a Rhombus is equal to one of its sides . Find the angles of the Rhombus

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$\large\underline{ \underline{ \sf \maltese\red{ \: Diagram:- }}}$

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\put(0,0){\line(1,3){1.5}}\put(0,0){\line(1,0){5}}\put(5,0){\line(1,3){1.5}}\put(1.5,4.5){\line(1,0){5}}\qbezier(1.56,4.5)(1.56,4.5)(5,0)\qbezier(6.45,4.5)(6.45,4.5)(0,0)\put(-0.5,-0.5){\sf B}\put(1,4.8){\sf A}\put(5.2,-0.5){\sf C}\put(6.7,4.75){\sf D}\put(3,1.6){\sf O}\end{picture}$

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$\large\underline{ \underline{ \sf \maltese{ \: Solution:- }}}$

Let ABCD be a Rhombus such that it's Diagonal BD is equal to its sides.

$\begin{gathered}\begin{gathered}\begin{gathered}: \implies \underline\blue{ \boxed{\displaystyle \sf \bold\orange{\: AB \: = \: BC \: = \: CD \: = \: AD \: = \: BD }} }\\ \\\end{gathered}\end{gathered}\end{gathered}$

$\qquad \quad {:} \longrightarrow \sf{\sf{∆ \: ABD \: and ∆ BCD \: are \: Equilateral }}$

$\qquad \quad {:}\longrightarrow\sf{\sf{ \angle A \: = \: \angle C \: = \: 60°}}$ ( Adjacent Angles of a Parallelogram are Supplementary )

Now,

Angle A + Angle B

$\qquad \quad {:} \longrightarrow \sf{\sf{\angle A \: + \: \angle B \: = \: 180° }}$

$\qquad \quad {:} \longrightarrow \sf{\sf{60° \: + \: \angle B \: = \: 180° }}$

$\qquad \quad {:} \longrightarrow \sf{\sf{\: \angle B \: = \: 180° \: - \: 60° }}$

$\qquad\quad {:} \longrightarrow \underline \red{\boxed{\sf{\angle B \: = \: 120° }}}$

Hence,

• Angle A = Angle C { 60° }
• Angle B = Angle D { 120° }

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$\bf \therefore \; \angle \;A = 60°$

$\bf \therefore \; \angle \;B = 120°$

$\bf \therefore \; \angle \;C = 60°$

$\bf \therefore \; \angle \;D = 120°$

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More For Knowledge:-

$\underbrace\red{\boxed{ \sf \blue{ Important \: Properties \: of \: Rhombus }}}$

• All sides of a rhombus are equal.
• The opposite angle of a rhombus are equal.
• The adjacent angle of a rhombus are supplementary.
• The diagonals of a Rhombus bisect each other at right angles.

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