1. **State the problem:** We need to find the size of angle $x$ in a figure where a large triangle is intersected by two parallel lines creating a smaller triangle inside it. Given angles are $85^\circ$, $42^\circ$, and $34^\circ$. 2. **Identify known facts and rules:** The sum of angles in any triangle is $180^\circ$. 3. **Analyze the smaller triangle:** The smaller triangle has angles $85^\circ$, $42^\circ$, and $x$. 4. **Use the triangle angle sum formula:** $$85^\circ + 42^\circ + x = 180^\circ$$ 5. **Calculate $x$:** $$x = 180^\circ - 85^\circ - 42^\circ$$ $$x = 180^\circ - 127^\circ$$ $$x = 53^\circ$$ 6. **Conclusion:** The size of angle $x$ is $53^\circ$.