Understanding Quadrilateral Angle Properties

The question asks us to identify the type of quadrilateral ABCD given its four interior angles: $\angle A = 47^\circ$, $\angle B = 66^\circ$, $\angle C = 133^\circ$, and $\angle D = 114^\circ$. To solve this, we need to check the properties of different types of quadrilaterals against the given angle measures.

First, let's confirm if the sum of the angles is $360^\circ$, which is true for any quadrilateral:

Sum = $\angle A + \angle B + \angle C + \angle D$

Sum = $47^\circ + 66^\circ + 133^\circ + 114^\circ = 360^\circ$. The angles form a valid quadrilateral.

Evaluating Quadrilateral Types

We will examine the properties associated with the given options:

• Parallelogram: In a parallelogram, opposite angles are equal ($\angle A = \angle C$ and $\angle B = \angle D$), and consecutive angles sum to $180^\circ$.
  • Check opposite angles: $\angle A = 47^\circ$ and $\angle C = 133^\circ$. These are not equal. $\angle B = 66^\circ$ and $\angle D = 114^\circ$. These are also not equal.
  • Therefore, ABCD is not a parallelogram.
• Rhombus: A rhombus is a special type of parallelogram with all sides equal. Since ABCD is not a parallelogram, it cannot be a rhombus.
• Trapezium: A trapezium has at least one pair of parallel sides. If sides are parallel, consecutive angles between them sum to $180^\circ$.
  • If AD || BC, then $\angle A + \angle B = 180^\circ$ and $\angle C + \angle D = 180^\circ$.
    • $47^\circ + 66^\circ = 113^\circ \neq 180^\circ$.
  • If AB || DC, then $\angle A + \angle D = 180^\circ$ and $\angle B + \angle C = 180^\circ$.
    • $47^\circ + 114^\circ = 161^\circ \neq 180^\circ$.
  • Since no pair of consecutive angles sums to $180^\circ$, ABCD is not a trapezium.
• Cyclic Quadrilateral: A cyclic quadrilateral is one whose vertices lie on a circle. A key property is that the sum of opposite angles is $180^\circ$.
  • Check opposite angles:
    • Pair 1: $\angle A + \angle C = 47^\circ + 133^\circ = 180^\circ$.
    • Pair 2: $\angle B + \angle D = 66^\circ + 114^\circ = 180^\circ$.
  • Since both pairs of opposite angles sum to $180^\circ$, the quadrilateral ABCD satisfies the condition for a cyclic quadrilateral.

Conclusion

Based on the analysis of the angle properties, the quadrilateral ABCD, with angles $47^\circ$, $66^\circ$, $133^\circ$, and $114^\circ$, is a cyclic quadrilateral because both pairs of its opposite angles sum to $180^\circ$.