To find the missing angle $\angle 1$, we can use two key pieces of information:

1. The angles inside a triangle sum to $180^\circ$.
2. The angle $\angle 1$ is an exterior angle, and the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

Steps:

• Given the interior angles of $63^\circ$ and $77^\circ$, the sum is: $63^\circ + 77^\circ = 140^\circ$
• The exterior angle $\angle 1$ is equal to this sum: $\angle 1 = 140^\circ$

Thus, $\angle 1 = 140^\circ$.

Would you like more details or further clarification? Here are 5 questions to consider:

1. What is the sum of all interior angles in any triangle?
2. How does the exterior angle theorem apply to other polygons?
3. Can a triangle have more than one exterior angle at a vertex?
4. How can we use this method to find missing angles in quadrilaterals?
5. What happens if the triangle is an obtuse or right triangle?

Tip: Always remember that the exterior angle of a triangle is the sum of the two opposite interior angles, which is a key property in many geometric proofs.