A diogonal of a rectangle is inclined to one side of the rectangle at `25^(@)`.
The acute angle between the diagonals is
a) 55 b) 50 c) 40 d)25

Options

A. `55^(@)`
B. `50^(@)`
C. `40^(@) `
D. `25^(@)`

Answer

B

Explanation

We know that, diagonals of a rectangle are equal in length.

`therefore" "AC=BD`
`rArr" "(1)/(2)AC=(1)/(2)BD" "["dividing both sides by 2"]`
`rArr" "OA=OB" "["since, O is the mid-point of Ac and BD"]`
`angle2=angle1" " ["angles opposite to equal sides are equal"]`
`=25^(@)`
`therefore" "angle3=angle1+angle2" "["exterior angle is equal to the sum of two opposite interior angles"]`
`= 25^(@) +25^(@)=50^(@)`
Hence, the acute angle between the diagonals is `50^(@)`.
Alternate Method
Given, in a rectangle ABCD,
`angleACD = 25^(@)`
`therefore" "angle CAB=25 ^(@)" "["alternate interior angles"]`
`now," "angleBCA=90^(@)-25^(@)=65^(@)= angleDAC" "["alternate interior angles"]`

In rectangle, diagonals bisect each other.
`therefore" "OD=OBandOA = OC`
So, `DeltaODC and DeltaOAB` are conguent. `" "`[by SSS congruence rule]
`therefore" "angleOBA=25^(@)=angleDCA" "["by CPCT rule"]`
Now, `angleAOD` is exterior angle of `DeltaAOB`.
`therefore" "angleAOD=angleOAB+angleOBA=25^(@)+25^(@)=50^(@)`