Question

In a cyclic quadrilateral ABCD, angle A = (2x + 4) °, angle B = (y + 3) °, angle C = (2y + 10) ° and angle D = (4x – 5) °. Find the smallest angle.

Answer 1

Hint: Before solving this question, let us know about Cyclic Quadrilateral first.
CYCLIC QUADRILATERAL: A cyclic quadrilateral is a quadrilateral whose all vertices lie on a single circle.

The sum of the opposite angles of a cyclic quadrilateral is always 180°.

Complete step by step answer:
As we know that, the sum of the opposite angles of a cyclic quadrilateral is always 180°.

Therefore, angle A + angle C = 180°
And, angle B + angle D = 180°
According to the question,
Angle A = (2x + 4) °
Angle B = (y + 3) °
Angle C = (2y + 10) °
Angle D = (4x – 5) °
Therefore, (2x + 4) + (2y + 10) = 180°
And (y + 3) + (4x – 5) = 180°
Let us solve (2x + 4) + (2y + 10) = 180° first.
\[\Rightarrow \] 2x + 4 + 2y + 10 = 180
\[\Rightarrow \]2x + 2y + 14 = 180
\[\Rightarrow \]2x + 2y = 166
\[\Rightarrow \]2x = 166 – 2y
\[\begin{align}
  & x=\dfrac{166-2y}{2} \\
 &\Rightarrow y=\dfrac{166-2x}{2} \\
\end{align}\]
Let us solve (y + 3) + (4x – 5) = 180° now.
\[\Rightarrow \]y + 3 + 4x – 5 = 180
\[\Rightarrow \]y + 4x – 2 = 180
\[\Rightarrow \]y + 4x = 182
\[\Rightarrow \]4x = 182 – y
\[x=\dfrac{182-y}{4}\]
\[\Rightarrow \]y = 182 – 4x
Let us now equate the values of ‘x’.
\[\dfrac{166-2y}{2}=\dfrac{182-y}{4}\]
\[\Rightarrow \]4 (166 – 2y) = 2 (182 – y)
\[\Rightarrow \]664 – 8y = 364 – 2y
\[\Rightarrow \]664 – 364 = -2y + 8y
\[\Rightarrow \]300 = 6y
\[\Rightarrow \]50 = y
Let us now equate the values of ‘y’.
\[\dfrac{166-2y}{2}\] = 182 – 4x
\[\Rightarrow \]166 – 2x = 2 (182 – 4x)
\[\Rightarrow \]166 – 2x = 364 – 8x
\[\Rightarrow \]-2x + 8x = 364 – 166
\[\Rightarrow \]6x = 198
\[\Rightarrow \]x = 33
Now, as we have calculated the values of ‘x’ and ‘y’, let us substitute the values of ‘x’ and ‘y’ in the measures of the angles of the quadrilateral ABCD.
Angle A = (2x + 4) ° = \[\left( 2\times 33+\text{ }4 \right)\]= (66 + 4) = 70°
Angle B = (y + 3) ° = (50 + 3) = 53°
Angle C = (2y + 10) ° = \[\left( 2\times 50+\text{ 10} \right)\] = (100 + 10) = 110°
Angle D = (4x – 5) ° = \[\left( 4\times 33\text{ }\text{ }5 \right)\] = (132 – 5) = 127°

So, the smallest angle, i.e. the angle with the smallest measure is Angle B, whose measure is 53°.

Note: Let us discuss the other properties of a cyclic quadrilateral.
The opposite angles of a cyclic quadrilateral are supplementary. In other words, the sum of the either pair of opposite angles of a cyclic quadrilateral is 180°.
If one side of a cyclic quadrilateral is produced, then the exterior angles will be equal to the opposite interior angle.
If the sum of any pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.