Question

If $ABCD$ is rhombus in which $\angle A={120}^o$ and $\overline{AC}$ is the diagonal, then $△ABC$ is __________.

• A. An obtuse angled triangle
• B. An isosceles triangle
• C. A scalene triangle
• D. An equilateral triangle

Answer 1

The correct option is D An equilateral triangle

$\Rightarrow$ In a given `figure ABCD is a rhombus.

$\Rightarrow$ $\angle$A = 120${}^{\circ }$ [Given]

$\Rightarrow$ Diagonal AC bisect $\angle$A in such a way that,

$\angle$CAB = 60${}^{\circ }$ ----- ( 1 )

$\Rightarrow$ AB = BC [All sides of rhombus are equal]

$\Rightarrow$ So, $\angle$CAB = $\angle$ACB [Angle of equal sides] --- ( 2 )

$\therefore$ $\angle$CAB = $\angle$ACB = 60${}^{\circ }$ [From (1) and (2)]

$\Rightarrow$ In $△$ABC,

$\Rightarrow$ $\angle$CAB + $\angle$ACB + $\angle$ABC = 180${}^{\circ }$

$\Rightarrow$ we get $\angle$ABC = 60${}^{\circ }$

$\Rightarrow$ So, $\angle ABC=\angle ACB=\angle CAB={60}^{\circ }$ ---- ( 3 )

Hence we get, $AB=BC=AC$ [From ( 3 ) ] --- ( 4 )

From ( 3 ) and ( 4 ) we have proved that,

$△ABCisanequilateraltriangle.$