Question

ABCD is a rhombus such that one of its diagonals is equal to its side. Find the measure of the angles of rhombus ABCD.

• A. ${45}^o,{135}^o,{45}^o,{135}^o$
• B. ${100}^o,{80}^o,{100}^o,{80}^o$
• C. ${120}^o,{60}^o,{120}^o,{60}^o$
• D. ${60}^o,{60}^o,{60}^o,{60}^o$

Answer 1

The correct option is C ${120}^o,{60}^o,{120}^o,{60}^o$

$\Rightarrow$ Let ABCD is a rhombus. AC and BD are diagonals such that, AC is equal to all sides of rhombus.

$\Rightarrow$ AB = BC = CD = DA = Diagonal AC.

$\Rightarrow$ In $△$ABC,

$\Rightarrow$ AB = BC = AC [Given]

$\therefore$ $△ABC$ is an equilateral triangle.

$\Rightarrow$ So, $\angle$CAB = $\angle$ABC = $\angle$ACB = 60${}^{\circ }$ [Angles of equilateral triangle] --- ( 1 )

$\Rightarrow$ Similarly, in $△$ADC we get,

$\Rightarrow$ $\angle$CAD = $\angle$ADC =$\angle$ACD = 60${}^{\circ }$ ----- ( 2 )

$\Rightarrow$ $\angle$BAD = $\angle$BAC + $\angle$CAD = 60${}^{\circ }$ + 60${}^{\circ }$ = 120${}^{\circ }$ -- ( 3 )

$\Rightarrow$ $\angle$BCD = $\angle$BCA + $\angle$ACD = 60${}^{\circ }$ + 60${}^{\circ }$ = 120${}^{\circ }$ ---- ( 4 )

So, from ( 1 ), ( 2 ), ( 3 ) and ( 4 ) we get,

The angles of a rhombus ABCD are ${120}^{\circ },{60}^{\circ },{120}^{\circ },{60}^{\circ }$.