From a point \( T \), a tangent \( T P \) at point \( P \), is drawn to a circle with centre \( O \). A secant TQR (point \( Q \) is near to point \( T \) ) is drawn from the point \( T \). \( \triangle P Q R \) is inscribed into the circle by joining the points \( P, Q \) and \( R \). Draw lines \( O Q \) and \( O R \). If \( \angle P T Q \) is \( 27^{\circ} \) and \( \angle T P Q= \) \( 55^{\circ} \), what is the degree measure of \( \angle R O Q \) ? 1.86 2.94

To find the measure of angle \( \angle R O Q \), we can use the properties of tangents and secants in relation to a circle. 1. **Identify the angles**: - We have \( \angle P T Q = 27^\circ \) and \( \angle T P Q = 55^\circ \). - Since \( T P \) is a tangent to the circle at point \( P \), we know that the angle between the tangent and the secant (or chord) is equal to the angle subtended by the chord at the opposite side of the circle. 2. **Calculate \( \angle P Q R \)**: - The angle \( \angle P Q R \) is the angle subtended by the chord \( P R \) at point \( Q \). - By the tangent-secant theorem, we have: \[ \angle P T Q = \angle P Q R \] - Therefore, \( \angle P Q R = 27^\circ \). 3. **Find \( \angle O Q P \)**: - The angle \( \angle O Q P \) is the angle at the center \( O \) subtended by the same arc \( P R \). - The angle at the center is twice the angle at the circumference subtended by the same arc: \[ \angle O Q P = 2 \times \angle P Q R = 2 \times 27^\circ = 54^\circ. \] 4. **Calculate \( \angle R O Q \)**: - Now, we can find \( \angle R O Q \) using the fact that the angles around point \( O \) sum to \( 180^\circ \): \[ \angle R O Q + \angle O Q P + \angle O P Q = 180^\circ. \] - We already know \( \angle O Q P = 54^\circ \) and \( \angle O P Q = 55^\circ \): \[ \angle R O Q + 54^\circ + 55^\circ = 180^\circ. \] - Simplifying this gives: \[ \angle R O Q + 109^\circ = 180^\circ, \] \[ \angle R O Q = 180^\circ - 109^\circ = 71^\circ. \] Thus, the measure of \( \angle R O Q \) is \( 71^\circ \). However, since the options provided are \( 1.86 \) and \( 2.94 \), it seems there might be a misunderstanding or miscommunication regarding the angles or the context of the problem. If the question is asking for a different angle or if there are additional constraints, please clarify. Based on the calculations, \( \angle R O Q = 71^\circ \).