The problem asks us to find the measure of angle 1 ($m\angle 1$) in a circle, where angle 1 is formed by the intersection of two chords. The intercepted arcs are $50^\circ$ and $56^\circ$.

1. Problem Description

The problem asks us to find the measure of angle 1 ($m\angle 1$) in a circle, where angle 1 is formed by the intersection of two chords. The intercepted arcs are $50^\circ$ and $56^\circ$.

2. Solution Steps

The measure of an angle formed by two chords intersecting inside a circle is equal to one-half the sum of the intercepted arcs.

The formula for finding the measure of the angle is:

$m\angle 1 = \frac{1}{2} (m\text{arc }LQ + m\text{arc }PR)$

In this problem, $m\text{arc }LQ = 50^\circ$ and $m\text{arc }PR = 56^\circ$. Plugging these values into the formula, we get:

$m\angle 1 = \frac{1}{2} (50^\circ + 56^\circ)$

$m\angle 1 = \frac{1}{2} (106^\circ)$

$m\angle 1 = 53^\circ$

3. Final Answer

53