In the diagram, $\triangle Q R S$ is an isosceles right-angled triangle with $Q R=S R$ and $\angle Q R S=90^{\circ}$. Line segment $P T$ intersects $S Q$ at $U$ and $S R$ at $V$. If $\angle P U Q=\angle R V T=y^{\circ}$, the value of $y$ is
(A) 72.5
(B) 60
(C) 67.5
(D) 62.5
(E) 70

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-016.jpg?height=355&width=311&top_left_y=1709&top_left_x=1292)
Since $\triangle Q R S$ is an isosceles right-angled triangle with $Q R=S R$, then $\angle R Q S=\angle R S Q=45^{\circ}$. Opposite angles are equal in measure, and so $\angle S U V=\angle P U Q=y^{\circ}$ and $\angle S V U=\angle R V T=y^{\circ}$. In $\triangle S V U, \angle V S U+\angle S U V+\angle S V U=180^{\circ}$ or $45^{\circ}+y^{\circ}+y^{\circ}=180^{\circ}$ or $2 y=135$ and so $y=67.5$.

ANSWER: (C)