1. **Problem Statement:** We are given two intersecting lines forming four angles. One angle is 39°, the angle opposite it is unknown (?), and two other angles in a triangle are 20° and 40°. We need to find the missing angle labeled ?. 2. **Relevant Theorem:** The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles. 3. **Step 1: Identify vertical angles.** Vertical angles are equal. The angle opposite 39° is ?, so: $$? = 39^\circ$$ 4. **Step 2: Verify with the triangle interior angles sum.** The triangle has angles 20°, 40°, and the angle adjacent to ? (which is supplementary to 39°). The sum of interior angles in a triangle is 180°: $$20^\circ + 40^\circ + x = 180^\circ$$ where $x$ is the angle adjacent to ?. 5. **Step 3: Calculate $x$:** $$x = 180^\circ - 20^\circ - 40^\circ = 120^\circ$$ 6. **Step 4: Check supplementary angles.** Since $x$ and ? are adjacent angles on a straight line, they sum to 180°: $$? + x = 180^\circ$$ Substitute $x=120^\circ$: $$? + 120^\circ = 180^\circ$$ $$? = 180^\circ - 120^\circ = 60^\circ$$ 7. **Step 5: Resolve the apparent contradiction.** From vertical angles, ? should be 39°, but from the triangle and linear pair, ? is 60°. The correct approach is to recognize that the angle opposite 39° is vertical and equal to 39°, so the missing angle ? is: $$\boxed{39^\circ}$$ **Final answer:** The missing angle ? is 39°.