Question

in the diagram, m∠lmn = 116, m∠jkm = 122, and m∠jnm = 103. what is m∠nkm? m∠nkm = (simplify your answer. do not include the degree symbol in your answer.)

Explanation:

Step1: Recall sum of angles in a quadrilateral

The sum of interior angles of a quadrilateral is 360°. In quadrilateral JNML, we know \(m\angle JNM = 103^{\circ}\), \(m\angle LMN=116^{\circ}\), and \(m\angle JKM = 122^{\circ}\). Let \(m\angle NK M=x\).

Step2: Set up the angle - sum equation

In quadrilateral JNML, \(m\angle JNM+m\angle NML + m\angle MLJ+m\angle LJN=360^{\circ}\). Also, considering the angles around point K, we can use the fact that the sum of angles in the figure. We know that the sum of angles in \(\triangle JNK\) and \(\triangle KML\) and the angles at the vertices of the overall figure. Another way is to note that if we consider the angles related to the given ones, we know that the sum of angles around the non - overlapping regions of the figure formed by the angles at N, M, and K.
We know that the sum of angles in the figure formed by the given angles and the unknown angle \(x\) can be written as:
\[m\angle JNM + m\angle LMN+m\angle JKM+x=360^{\circ}\]

Step3: Substitute the given values

Substitute \(m\angle JNM = 103\), \(m\angle LMN = 116\), and \(m\angle JKM = 122\) into the equation:
\[103+116 + 122+x=360\]
\[341+x=360\]

Step4: Solve for \(x\)

Subtract 341 from both sides of the equation:
\[x=360 - 341\]
\[x = 19\]

Answer:

19