The angles of a cyclic quadrilateral, taken in order are x°, (3x - 30)°, (y + 30)° and (2x - y)°. Find the measure of the smallest angle of the quadrilateral.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 30°
In a cyclic quadrilateral the opposite angles add up to 180°. Taken in order, the first and third angles are opposite, and so are the second and fourth.
The two equations:
\(x+(y+30)=180\), so \(x+y=150\) (1)
\((3x-30)+(2x-y)=180\), so \(5x-y=210\) (2)
Adding (1) and (2), the \(y\) terms cancel: \(6x=360\), so \(x=60\)
Putting \(x=60\) in (1): \(y=90\)
The four angles:
\(x=60^\circ\) and \(3x-30=150^\circ\)
\(y+30=120^\circ\) and \(2x-y=30^\circ\)
The smallest angle is 30°.