\( P \) is a point on \( A B \). \( Q \) is a point on \( A C \). \( A P=A Q \). Work out the size of angle \( A Q P \). You must give a reason for each stage of your working.

To determine the size of angle \( A Q P \), we first note that since \( A P = A Q \), triangles \( A P Q \) is an isosceles triangle. This means that angles opposite these sides are equal. Consequently, angle \( A P Q = A Q P \). Next, we can use the fact that the sum of angles in triangle \( A P Q \) must equal \( 180^\circ \). Hence, if we let \( x \) represent the measure of angles \( A P Q \) and \( A Q P \), we have: \[ x + x + \text{angle} A = 180^\circ \] This simplifies to: \[ 2x + \text{angle} A = 180^\circ \] From this equation, we can solve for \( x \) (which is the measure of angle \( A Q P \)) by rearranging: \[ 2x = 180^\circ - \text{angle} A \] \[ x = \frac{180^\circ - \text{angle} A}{2} \] This means angle \( A Q P \) is determined based on the measure of angle \( A \). Thus, any specific numerical value for angle \( A \) will lead to a specific size for angle \( A Q P \) using the equation above. In conclusion, angle \( A Q P \) is equal to \( \frac{180^\circ - \text{angle} A}{2} \), due to the property of isosceles triangles and the angle sum property.