+0

# help geometry

0
2
1
+1121

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 $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, then what is $\angle ZNM + \angle ZMN + \angle PNM + \angle PMN + \angle NZM + \angle NPM$ in degrees?

Apr 15, 2024

#1
+179
0

Since ∠TSU=60∘ and ∠STU=45∘, then ∠SUT=75∘. Triangle STU is a 45-45-90 triangle, so SU=UT=2​⋅ST. Also, since M and N are midpoints, then SM=MT=NU=UT=2​⋅ST and TN=NS=US=2​⋅ST.

Triangle UMN is isosceles with UM=MN because they are both half of sides of an isosceles triangle. Similarly, triangle PNM is isosceles with PN=MN. Since ∠SUT=75∘, then ∠SUM=∠SUN=(180∘−75∘)/2=52.5∘. Likewise, ∠PNM=∠PMN=52.5∘.

Looking at triangle ZNM, we see that ∠ZNM+∠ZMN+∠MZN=180∘. Since ∠MZN is a right angle, then ∠ZNM+∠ZMN=90∘. Similarly, for triangle PNM, we have ∠PNM+∠PMN=90∘.

Adding all the angle measures together, we get the answer: $\angle ZNM + \angle ZMN + \angle PNM + \angle PMN + \angle NZM + \angle NPM = 90^\circ + 90^\circ + 52.5^\circ + 52.5^\circ + \angle NZM + \angle NPM.$

Since ∠NZM and ∠NPM are supplementary angles (they form a straight line), then ∠NZM+∠NPM=180∘. Therefore, the final answer is $90^\circ + 90^\circ + 52.5^\circ + 52.5^\circ + 180^\circ = \boxed{465^\circ}$.

Apr 15, 2024