Question

first, find ( mangle jqk ).
( mangle jqk = 70 )
now, find ( mangle mqn ).
( mangle mqn = )

Explanation:

Step1: Use the property of angles around a point

The sum of angles around a point \(Q\) is \(360^{\circ}\). But we can also use the fact that vertical - angle pairs and linear - angle pairs. Another way is to note that the sum of angles in a full - circle around \(Q\) can be considered by looking at the non - overlapping angles. We know that \(m\angle JQK = 70^{\circ}\), \(m\angle KQL=30^{\circ}\), \(m\angle QPN = 80^{\circ}\) (vertically opposite to the given \(80^{\circ}\) angle).
Since the sum of angles around a point \(Q\) is \(360^{\circ}\), and we can also use the fact that the sum of adjacent angles forming a full - circle can be calculated as \(m\angle JQK+m\angle KQL + m\angle LQM+m\angle MQN+m\angle NQP+m\angle PJQ=360^{\circ}\). But a simpler approach is to use the fact that the sum of angles in a circle is \(360^{\circ}\) and we can group the angles. Also, we know that \(m\angle JQK\) and \(m\angle LQM\) are vertical angles (if we consider the intersection of lines), \(m\angle KQL\) and \(m\angle NQP\) are vertical angles, and \(m\angle MQN\) and \(m\angle PJQ\) are vertical angles. But another way is to use the sum of angles in a circle: \(m\angle JQK + m\angle KQL+m\angle MQN+80^{\circ}=180^{\circ}\) (because they form a linear - pair of angles when considering the straight - line concept in the circle of angles around \(Q\)).

Step2: Solve for \(m\angle MQN\)

We know that \(m\angle JQK = 70^{\circ}\) and \(m\angle KQL = 30^{\circ}\). Substitute these values into the equation \(m\angle JQK+m\angle KQL+m\angle MQN=180^{\circ}- 80^{\circ}\) (sum of angles on a straight - line).

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Answer:

\(80\)