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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?

 Mar 15, 2024
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\(\because \triangle TSU \sim \triangle TPM\)

\(\therefore \angle TMP = \angle TUS = {89}^{\circ}\)

\(\because \angle STU = {29}^{\circ}, \angle TZU = {90}^{\circ}\)

\(\therefore \angle TUZ = 180-29-90={61}^{\circ}\)

\(\therefore \angle TMZ + \angle TUZ = {160}^{\circ}\).

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 Mar 16, 2024

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