1. **Problem Statement:** We are given triangle MTR with point T above segment MR. The triangle has two equal sides: MT = TR, and angles at T are 35° and 25° on either side. Points N, P, Q lie on MR with MN = 4, PQ = 6, and P is the foot of the perpendicular from T to MR. We need to find the length of MR. 2. **Understanding the problem:** Since MT = TR, triangle MTR is isosceles with T as the apex. The angles at T split into 35° and 25°, so the total angle at T is 60°. 3. **Key formula:** In an isosceles triangle with equal sides $MT = TR = s$ and base $MR = b$, the altitude from T to MR bisects MR and forms two right triangles. The altitude $h$ satisfies: $$h = s \sin(30^\circ)$$ and half the base is: $$\frac{b}{2} = s \cos(30^\circ)$$ 4. **Given data:** The base MR is divided into segments MN = 4, NP (height foot), PQ = 6, and QR unknown. Since P is the foot of the perpendicular from T, and the altitude bisects MR, then: $$MN + NP + PQ + QR = MR$$ 5. **Using the altitude and isosceles property:** The altitude bisects MR, so: $$MN + NP + PQ + QR = 2(MN + NP)$$ 6. **Calculate MR:** Since MN = 4 and PQ = 6, and P is midpoint, then: $$MR = 2(MN + NP) = 2(4 + 6) = 20$$ 7. **Final answer:** The length of MR is: $$\boxed{20}$$