# Problem 9.3

In a convex pentagon $P Q R S T$, angle $P R T$ is half the size of angle $Q R S$, and all sides are equal. Find angle $P R T$.

## Number of points 7
Answer:

$30^{\circ}$.

Solution. From the condition of the problem, it follows that $\angle P R Q+\angle T R S=\angle P R T(*)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f1e15e56af4a180d9426g-2.jpg?height=591&width=562&top_left_y=1195&top_left_x=290)

Figure a

![](https://cdn.mathpix.com/cropped/2024_05_06_f1e15e56af4a180d9426g-2.jpg?height=597&width=417&top_left_y=1198&top_left_x=1116)

Figure b

First method. We use the "folding" method. Reflect triangle $P Q R$ symmetrically with respect to line $P R$, and triangle $T S R$ with respect to line $T R$ (see Fig. a). From the equality (*) and the equality $R Q=R S$, it follows that the images of points $Q$ and $S$ are the same point $O$.

Note that triangle $T O P$ is equilateral. Moreover, $O R=O P=O T$. Therefore, $O$ is the center of the circumscribed circle of triangle $P R T$. Then $\angle P R T=$ $0.5 \angle P O T=30^{\circ}$.

Second method. We prove that QPTS is a parallelogram (see Fig. b).

Indeed, using the equality of the angles at the bases of the isosceles triangles $P Q R$ and $R S T$ and the equality (*), we get: $\angle Q P T+\angle P T S=\angle Q P R+\angle R P T+$ $\angle R T P+\angle S T R=\angle P R Q+\angle T R S+\left(180^{\circ}-\angle P R T\right)=180^{\circ}$.

Thus, $P Q \| S T$ and $P Q=S T$ (by condition), that is, $Q P T S$ is a parallelogram. Then $Q S=P T$, so triangle $Q R S$ is equilateral.

Therefore, $\angle P R T=0.5 \angle Q R S=30^{\circ}$.

## Additional grading criteria

"7" A complete and justified solution is provided

“4-6" A generally correct solution is provided, containing minor gaps or inaccuracies (for example, the fact that the images of points $\mathrm{Q}$ and $\mathrm{S}$ coincide under the symmetries is used but not justified)

"1-3" The correct answer is obtained, assuming that QPTS is a parallelogram, but this is not proven

"0" Only the answer is provided or the answer is obtained by considering a regular pentagon

" 0 " An incorrect solution is provided or no solution is provided