In the diagram N, R and S are points on a circle. MP is a tangent to the circle at N. If |NR| = |RS| and SNP = 74°, find RNS.

The angle between the tangent $MP$ and the chord $NS$ at the point of contact $N$ is equal to the angle subtended by the chord $NS$ in the alternate segment.

$$\angle NRS = \angle SNP$$

Given $\angle SNP = 74^\circ$, so:

$$\angle NRS = 74^\circ$$

We are given that $|NR| = |RS|$. This means that triangle $\triangle NRS$ is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal.
Therefore, the angle opposite side $RS$ ($\angle RNS$) is equal to the angle opposite side $NR$ ($\angle NSR$).

$$\angle RNS = \angle NSR$$

The sum of angles in any triangle is $180^\circ$. For $\triangle NRS$:

$$\angle NRS + \angle RNS + \angle NSR = 180^\circ$$

Substitute the values from Step 1 and Step 2:

$$74^\circ + \angle RNS + \angle RNS = 180^\circ$$ $$74^\circ + 2 \angle RNS = 180^\circ$$

Subtract $74^\circ$ from both sides:

$$2 \angle RNS = 180^\circ - 74^\circ$$ $$2 \angle RNS = 106^\circ$$

Divide by 2:

$$\angle RNS = \frac{106^\circ}{2}$$ $$\angle RNS = \boxed{\text{53}^\circ}$$