In Quadrilateral ABCD, + = 140^ , : = 1 : 3 and : = 5 : 6. Find the values of , , and ?

In Quadrilateral ABCD, + = 140^ , : = 1 : 3 and : = 5 : 6. Find t…

MCQ

In Quadrilateral $ABCD$, $\angle\text{A}+ \angle\text{C} = 140^\circ, \ \angle\text{A} : \angle\text{C} = 1 : 3$ and $\angle\text{B} : \angle\text{D} = 5 : 6.$ Find the values of $\angle\text{A}, \ \angle\text{B},\ \angle\text{C}$ and $\angle\text{D}?$

A
$10^\circ , 20^\circ , 100^\circ , 260^\circ $
✓
$35^\circ , 100^\circ , 105^\circ , 120^\circ $
C
$100^\circ , 102^\circ , 120^\circ , 10^\circ $
D
$90^\circ , 90^\circ , 100^\circ , 80^\circ $

✓

Answer

Correct option: B.

$35^\circ , 100^\circ , 105^\circ , 120^\circ $

given: $\angle\text{A}+ \angle\text{C}=140^\circ$
and $\angle\text{A}:\angle\text{C} = 1:3$
and $\angle\text{B}:\angle\text{D} = 5:6$
$\Rightarrow \angle\text{A}= \frac{1}{4} \times 140^\circ - 35^\circ$
$\Rightarrow \angle\text{C}= \frac{3}{4} \times 140^\circ - 105^\circ$
Now according to angle sum property of quadrilateral
$\angle\text{A} +\angle\text{B}+ \angle\text{C}+ \angle\text{D} = 360^\circ$
$\Rightarrow 35^\circ+ \angle\text{B}+ 105^\circ +\angle\text{D} = 360^\circ$
$\Rightarrow \angle\text{B}+ \angle\text{D} = 360^\circ - 140^\circ = 220^\circ$
$\Rightarrow 5\text{x} + 6\text{x} = 220^\circ$
$\Rightarrow\text{x}=20^\circ$
So, $\angle\text{B} = 5 \times 20^\circ = 100^\circ$
and $\angle\text{D} = 6 \times 20^\circ = 120^\circ$