Question

In a cyclic quadrilateral \(ABCD\), what is \(\angle ABC\)? Statement (I): \(\angle ABC + \angle BCD = 125^\circ\) Statement (II): \(\angle ADC = 125^\circ\)

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

Whenever the words “cyclic quadrilateral” appear, immediately recall that opposite angles add up to \(180^\circ\).

Answer 1

The Correct Option is B

Concept: A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. The most important property of a cyclic quadrilateral is: \[ \text{Opposite angles are supplementary.} \] Therefore, \[ \angle ABC+\angle ADC=180^\circ \] and \[ \angle BAD+\angle BCD=180^\circ. \] This property is frequently used to determine unknown angles.

Step 1: Analyze Statement (I). Given: \[ \angle ABC+\angle BCD=125^\circ. \] This relation involves two adjacent angles. No individual value of either angle is known. For example, \[ \angle ABC=60^\circ,\quad \angle BCD=65^\circ \] satisfies the condition. Also, \[ \angle ABC=70^\circ,\quad \angle BCD=55^\circ \] satisfies the same condition. Hence infinitely many values of \(\angle ABC\) are possible. Therefore Statement (I) alone is not sufficient.

Step 2: Analyze Statement (II). Given: \[ \angle ADC=125^\circ. \] Since \(ABCD\) is cyclic, \[ \angle ABC+\angle ADC=180^\circ. \] Substituting: \[ \angle ABC+125^\circ=180^\circ \] \[ \angle ABC=55^\circ. \] A unique value is obtained. Therefore Statement (II) alone is sufficient.