Question : In a quadrilateral ABCD, E is a point in the interior of the quadrilateral such that DE and CE are the bisectors of $\angle D$ and $\angle C$, respectively. If $\angle B=82^{\circ}$ and $\angle D E C=80^{\circ}$, then $\angle A=$ ?
Option 1: 75º
Option 2: 81º
Option 3: 84º
Option 4: 78º
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Correct Answer: 78º
Solution :
Concept used:
The sum of all the internal angles of a quadrilateral is 360$^\circ$
The sum of all the internal angles of a triangle is 180$^\circ$
Calculations:
In quadrilateral ABCD,
$\angle$A + $\angle$B + $\angle$C + $\angle$D = 360$^\circ$
⇒ $\angle$A + 82° + $\angle$C + $\angle$D = 360$^\circ$
⇒ $\angle$A + $\angle$C + $\angle$D = 278$^\circ$ ----(1)
In $\triangle$DEC,
$\angle$EDC + $\angle$DEC + $\angle$ECD = 180$^\circ$
⇒ $\frac{\angle \text{D}}{2}$ + 80$^\circ$ + $\frac{\angle \text{C}}{2}$ = 180$^\circ$
⇒ $\frac{\angle \text{D}}{2}$ + $\frac{\angle \text{C}}{2}$ = 100$^\circ$
⇒ $\angle$D + $\angle$C = 200$^\circ$ ----(2)
Put the value of $\angle$D + $\angle$C from eq.(2) to eq. (1)
⇒ $\angle$A + 200$^\circ$ = 278$^\circ$
⇒ $\angle$A = 78$^\circ$
$\therefore$ The value of $\angle$A is 78$^\circ$
Hence, the correct answer is 78$^\circ$.
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