The five sides of a regular pentagon are all equal in length. Also, all interior angles of a regular pentagon have the same measure. In the diagram, $P Q R S T$ is a regular pentagon and $\triangle P U T$ is equilateral. The measure of obtuse $\angle Q U S$ is
(A) $172^{\circ}$
(B) $168^{\circ}$
(C) $170^{\circ}$
(D) $176^{\circ}$
(E) $174^{\circ}$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-028.jpg?height=382&width=396&top_left_y=999&top_left_x=1344)
Join $Q U$ and $S U$.

Since $\triangle P U T$ is equilateral, then $P U=U T=T P$.

Since pentagon $P Q R S T$ is regular, then $Q P=P T=T S$.

Thus, $P U=Q P$ and $U T=T S$, which means that $\triangle Q P U$ and $\triangle S T U$ are isosceles.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-054.jpg?height=374&width=398&top_left_y=718&top_left_x=923)

Each interior angle in a regular pentagon measures $108^{\circ}$.

Since $\angle U P T=60^{\circ}$, then $\angle Q P U=\angle Q P T-\angle U P T=108^{\circ}-60^{\circ}=48^{\circ}$.

Since $\triangle Q P U$ is isosceles with $Q P=P U$, then $\angle P Q U=\angle P U Q$.

Thus, $\angle P U Q=\frac{1}{2}\left(180^{\circ}-\angle Q P U\right)=\frac{1}{2}\left(180^{\circ}-48^{\circ}\right)=66^{\circ}$.

By symmetry, $\angle T U S=66^{\circ}$.

Finally, $\angle Q U S=360^{\circ}-\angle P U Q-\angle P U T-\angle T U S=360^{\circ}-66^{\circ}-60^{\circ}-66^{\circ}=168^{\circ}$.

ANSWER: (B)