1. **State the problem:** We have a large triangle divided into two smaller triangles by a segment of equal length in both triangles. The left smaller triangle has angles 35° and 70°, and the right smaller triangle has an angle labeled $x$. We need to find the value of $x$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always 180°. 3. **Analyze the left smaller triangle:** It has angles 35° and 70°, so the third angle is $$180^\circ - 35^\circ - 70^\circ = 75^\circ.$$ 4. **Equal segments imply isosceles triangles:** The segment dividing the large triangle is equal in length in both smaller triangles, and the tick marks indicate the sides opposite to these angles are equal. This suggests the right smaller triangle is isosceles with two equal sides. 5. **Use the isosceles triangle property:** In the right smaller triangle, the two equal sides imply the two base angles are equal. One angle is $x$, so the other base angle is also $x$. 6. **Sum of angles in the right smaller triangle:** Let the third angle be $y$. Then $$2x + y = 180^\circ.$$ 7. **Relate the angles between the two triangles:** The angle adjacent to the 75° angle in the large triangle is $y$, and since the large triangle is composed of these two smaller triangles, the sum of angles around the point where the two smaller triangles meet must be 180°. 8. **Calculate $y$:** The angle adjacent to 75° is supplementary to 75°, so $$y = 180^\circ - 75^\circ = 105^\circ.$$ 9. **Solve for $x$:** Substitute $y=105^\circ$ into the equation $$2x + 105^\circ = 180^\circ,$$ $$2x = 180^\circ - 105^\circ = 75^\circ,$$ $$x = \frac{75^\circ}{2} = 37.5^\circ.$$ **Final answer:** $$x = 37.5^\circ.$$