In $\Delta ABC$,$m \angle A = x$,$m \angle B = 3x$,and $m \angle C = y$. If $3y - 5x = 30$,show that the triangle is right-angled and find the measures of all three angles.

(A) The sum of angles in a triangle is $180^{\circ}$. Therefore,$x + 3x + y = 180$,which simplifies to $4x + y = 180$.
From this,we get $y = 180 - 4x$.
Substituting this into the given equation $3y - 5x = 30$:
$3(180 - 4x) - 5x = 30$
$540 - 12x - 5x = 30$
$540 - 17x = 30$
$17x = 510$
$x = 30^{\circ}$.
Now,find $y$:
$y = 180 - 4(30) = 180 - 120 = 60^{\circ}$.
Wait,checking the angles: $m \angle A = 30^{\circ}$,$m \angle B = 3(30) = 90^{\circ}$,and $m \angle C = 60^{\circ}$.
Since one angle is $90^{\circ}$,the triangle is right-angled.
The angles are $30^{\circ}, 90^{\circ}, 60^{\circ}$.