Question

one of the diagonals of a rhombus is equal to one of its sides,. find the angle of the rhombus.

Answer 1

Given:  ABCD is a rhombus.

Let DX is the altitude from D to AB

Then AX = BX (DX bisects AB)

 

Now in ΔAXD and ΔBXD

AX = BX

∠AXD = ∠BXD = 90° (DX is altitude)

DX = DX (Common)

Thus ΔAXD congruent to ΔBXD (by RHS congruency criterion)

⇒ ∠DAX = ∠DBX

⇒ ∠DAB = ∠DBA ... (1)

 

but diagonal of a rhombus bisects the angles

⇒ ∠CBA = 2∠DBA ... (2)

 

from (1) and (2) we get

∠CBA = 2∠DAB

 

we know that adjacent angles of a rhombus are supplementary

⇒ ∠DAB + ∠CBA = 180°

⇒ ∠DAB + 2∠DBA = 180°

⇒ 3∠DAB = 180°

⇒ ∠DAB = 180°/3

⇒ ∠DAB = 60°

and ∠CBA = 2 × 60° = 120°

 

also the opposite angles of rhombus are equal

⇒ ∠BCD = ∠DAB = 60°

and ∠ADC = ∠CBA = 120°

 

Hence

∠A = 60°, ∠B = 120°, ∠C = 60° and ∠D = 120