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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.

Find the missing angle (∠1) given that the interior angles are 63° and 77°.

Learn how to find the missing angle (∠1) in a triangle where the given angles are 63° and 77°. This geometry problem covers the concept of the sum of interior angles and uses the Exterior Angle Theorem to find the solution. Suitable for students in Grades 6-8.

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