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Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively.  Let $\overline{UZ}$ be an altitude of the triangle.  If $\angle TSU = 62^\circ$ and $\angle STU = 29^\circ$, then what is $\angle TMP + \angle TUZ$ in degrees?

 Apr 30, 2024
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Note that \(\triangle TMP \sim \triangle TUS\). Then \(\angle TMP = \angle TUS = 180^\circ - \angle TSU - \angle STU = 89^\circ\).

 

Considering the interior angle sum in triangle TUZ gives 

\(90^\circ + \angle STU + \angle TUZ = 180^\circ\\ \angle TUZ = 90^\circ - \angle STU = 61^\circ\)

 

Therefore,

\(\angle TMP + \angle TUZ = 89^\circ + 61^\circ = \boxed{150^\circ}\)

 Apr 30, 2024

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