Question

first, find ( mangle jqk ).
( mangle jqk=square^{circ} )

Explanation:

Step1: Use the property of vertical angles

Vertical angles are equal. The angle opposite to \(m\angle JQK\) can be found by using the fact that the sum of angles around a point is \(360^{\circ}\). But another approach: if we assume that the lines are intersecting and we use the property of linear pairs and vertical angles. Here, we can note that \(m\angle JQK\) and the angle composed of \(80^{\circ}\) and \(30^{\circ}\) (through vertical - angle relationships). Wait, no. Let's use the property that the sum of angles around a point \(Q\) is \(360^{\circ}\). But more simply, if we consider the fact that when two lines intersect, vertical angles are equal. However, a better way: assume that the lines form pairs of vertical angles. Let's use the fact that the sum of adjacent angles on a straight - line - like arrangement (but in a circular \(360^{\circ}\) around \(Q\)). But actually, if we consider the vertical - angle relationship directly. The angle \(m\angle JQK\) and the angle opposite to it (if we consider the intersection of lines). Wait, another approach:
We know that the sum of angles around a point \(Q\) is \(360^{\circ}\). But if we assume that the figure has pairs of vertical angles. Let's assume that the non - overlapping angles around \(Q\): if we consider that \(m\angle JQK\) and the angle composed of \(80^{\circ}\) and \(30^{\circ}\) (through vertical - angle relationships). Wait, no. Let's use the property of vertical angles.
Let's assume that the two pairs of intersecting lines. The angle \(m\angle JQK\) and the angle \(70^{\circ}\) (because \(180-(80 + 30)=70\) in a linear - pair - like consideration for the full \(180^{\circ}\) straight - line - adjacent angles). Wait, no. Let's use the property that the sum of angles around a point \(Q\) is \(360^{\circ}\). But if we consider that the lines form two pairs of vertical angles. Let's assume that for one pair of intersecting lines, we have angles. But a more straightforward way:
We know that \(m\angle JQK=70^{\circ}\) because if we consider the fact that in the intersection of lines (assuming that the \(80^{\circ}\) and \(30^{\circ}\) angles are adjacent to \(m\angle JQK\) in a non - overlapping way around a point \(Q\) and using the \(180^{\circ}\) straight - line (semi - circular) angle sum.
Let's assume that there is a straight - line (semi - circle) of \(180^{\circ}\). If we have two angles \(80^{\circ}\) and \(30^{\circ}\) adjacent to \(m\angle JQK\) on a semi - circle. Then \(m\angle JQK=180-(80 + 30)\).

Step2: Calculate the value

$$m\angle JQK=180-(80 + 30)=180 - 110=70$$

Answer:

\(70\)