1. **State the problem:** We need to find the measure of angle $\angle MNT$ in the given triangle configuration. 2. **Analyze the diagram and given information:** - Triangle $MTR$ with vertex $T$ at the top. - Points $N$, $P$, and $Q$ lie on segment $MR$. - Segment $MN = 4$ units, segment $PQ = 6$ units. - Angle $T$ is split into two parts: $35^\circ$ and $25^\circ$. - There is a right angle at point $P$ between $NP$ and $TP$. - Tick marks indicate congruent segments on $MT$, $TN$, $TR$, and $TQ$. 3. **Identify relevant properties and formulas:** - The sum of angles in a triangle is $180^\circ$. - Right angle at $P$ means $\angle NPT = 90^\circ$. - Congruent segments imply isosceles triangles or equal lengths. 4. **Use angle addition at vertex $T$:** $$\angle MTR = 35^\circ + 25^\circ = 60^\circ$$ 5. **Consider triangle $MNT$:** - Since $TN$ is congruent to $MT$ (tick marks), triangle $MNT$ is isosceles with $MT = TN$. - Therefore, angles opposite these sides are equal: $\angle MNT = \angle NMT$. 6. **Calculate $\angle MNT$:** - Sum of angles in triangle $MNT$ is $180^\circ$. - Let $\angle MNT = \angle NMT = x$. - The third angle is $\angle MTN$, which is part of $\angle MTR$ and equals $60^\circ$. So, $$x + x + 60^\circ = 180^\circ$$ $$2x = 120^\circ$$ $$x = 60^\circ$$ 7. **Final answer:** $$\boxed{60^\circ}$$ is the measure of $\angle MNT$.