Question : $\angle A, \angle B$ and $\angle C$ are three angles of a triangle. If $\angle A- \angle B=15^{\circ}, \angle B - \angle C=30^{\circ}$, then $\angle A, \angle B$ and $\angle C$ are:
Option 1: $80^{\circ},60^{\circ},$ and $40^{\circ}$
Option 2: $70^{\circ},50^{\circ},$ and $60^{\circ}$
Option 3: $80^{\circ},65^{\circ},$ and $35^{\circ}$
Option 4: $80^{\circ},55^{\circ},$ and $45^{\circ}$
Correct Answer: $80^{\circ},65^{\circ},$ and $35^{\circ}$
Solution :
Given that $\angle A - \angle B = 15^{\circ}$ and $\angle B - \angle C = 30^{\circ}$.
$\angle A = \angle B + 15^{\circ}$
$\angle C = \angle B - 30^{\circ}$
We know that the sum of the angles in a triangle is $180^{\circ}$.
$\angle A + \angle B + \angle C = 180^{\circ}$
Substituting the expressions for $\angle A$ and $\angle C$.
$(\angle B + 15^{\circ}) + \angle B + (\angle B - 30^{\circ}) = 180^{\circ}$
⇒ $3\angle B - 15^{\circ} = 180^{\circ}$
⇒ $3\angle B = 195^{\circ}$
⇒ $\angle B = 65^{\circ}$
Substituting $\angle B = 65^{\circ}$ into the expressions for $\angle A$ and $\angle C$,
$\angle A = 65^{\circ} + 15^{\circ} = 80^{\circ}$
$\angle C = 65^{\circ} - 30^{\circ} = 35^{\circ}$
Hence, the correct answer is $80^{\circ}$, $65^{\circ}$, and $35^{\circ}$.
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