In a DABC, points P , Q and R are taken on AB, BC and CA, respectively, such that BQ = PQ and QC = QR. If ∠BAC = 75°, what is the measure of ∠PQR (in degrees)?

To find the measure of $\angle \mathrm{PQR}$, we can use the properties of triangles and angles on a straight line.

Let the angles of $△\mathrm{ABC}$ be:
$\angle \mathrm{BAC}={75}^{\circ }$
$\angle \mathrm{ABC}=\angle B$
$\angle \mathrm{ACB}=\angle C$

In $△\mathrm{ABC}$, the sum of the angles is ${180}^{\circ }$:
$\angle A+\angle B+\angle C={180}^{\circ }$
Substituting $\angle A={75}^{\circ }$:
${75}^{\circ }+\angle B+\angle C={180}^{\circ }$
$\angle B+\angle C={180}^{\circ }-{75}^{\circ }={105}^{\circ }$

Now, consider the points on the sides of the triangle:
1. In $△\mathrm{PBQ}$, we are given that $\mathrm{BQ}=\mathrm{PQ}$. Therefore, $△\mathrm{PBQ}$ is an isosceles triangle with angles opposite to equal sides being equal:
$\angle \mathrm{QPB}=\angle \mathrm{PBQ}=\angle B$
The sum of angles in $△\mathrm{PBQ}$ is ${180}^{\circ }$, so the angle at vertex $Q$ is:
$\angle \mathrm{PQB}={180}^{\circ }-2\angle B$

2. In $△\mathrm{RQC}$, we are given that $\mathrm{QC}=\mathrm{QR}$. Therefore, $△\mathrm{RQC}$ is also an isosceles triangle:
$\angle \mathrm{QRC}=\angle \mathrm{RCQ}=\angle C$
The angle at vertex $Q$ is:
$\angle \mathrm{RQC}={180}^{\circ }-2\angle C$

Since the points $B$, $Q$, and $C$ lie on a straight line, the sum of the angles at point $Q$ is ${180}^{\circ }$:
$\angle \mathrm{PQB}+\angle \mathrm{PQR}+\angle \mathrm{RQC}={180}^{\circ }$

Substitute the expressions we found for $\angle \mathrm{PQB}$ and $\angle \mathrm{RQC}$ into the equation:
$({180}^{\circ }-2\angle B)+\angle \mathrm{PQR}+({180}^{\circ }-2\angle C)={180}^{\circ }$
${360}^{\circ }-2(\angle B+\angle C)+\angle \mathrm{PQR}={180}^{\circ }$

Now substitute $\angle B+\angle C={105}^{\circ }$ into this equation:
${360}^{\circ }-2\left({105}^{\circ }\right)+\angle \mathrm{PQR}={180}^{\circ }$
${360}^{\circ }-{210}^{\circ }+\angle \mathrm{PQR}={180}^{\circ }$
${150}^{\circ }+\angle \mathrm{PQR}={180}^{\circ }$
$\angle \mathrm{PQR}={180}^{\circ }-{150}^{\circ }={30}^{\circ }$

Thus, the measure of $\angle \mathrm{PQR}$ is ${30}^{\circ }$.