In the given figure, MNP, SQP, NQR and MSR are straight lines. $\angle NPQ = 54^{\circ}$ and = $\angle QRS = 68^{\circ}$. What is the degree measure of $\angle$SMN?

$29^{\circ}$

In \(\Delta \)RMN,

= \(\theta \) + \(\alpha \) + \({68}^\circ\) = \({180}^\circ\)

= \(\theta \) + \(\alpha \) = \({180}^\circ\) - \({68}^\circ\) = \({112}^\circ\)

= \(\theta \) + \(\alpha \) = \({112}^\circ\)

In \(\Delta \)PMS,

= \(\theta \) + \({180}^\circ\) - \(\alpha \) + \({54}^\circ\) = \({180}^\circ\)

= \(\theta \) + \({54}^\circ\) = \(\alpha \)

Putting this value in above equation, we get,

= \(\theta \) + \(\theta \) + \({54}^\circ\) = \({112}^\circ\)

= 2\(\theta \) = \({112}^\circ\) - \({54}^\circ\) = \({58}^\circ\)

= \(\theta \) = \(\frac{58}{2}\) = \({29}^\circ\)

Therefore, \(\angle\)SMN is \({29}^\circ\).