The given question asks to find the measure of angle $\angle 1$ in the triangle, where:

• $\alpha = 35^\circ$
• $\beta = 89^\circ$

From the diagram, the three angles in the triangle formed by $\alpha$, $\beta$, and $\angle 1$ must sum to 180° (since the sum of angles in a triangle is always 180°). Therefore, we can set up the equation:

$\alpha + \beta + \angle 1 = 180^\circ$

Substituting the values for $\alpha$ and $\beta$:

$35^\circ + 89^\circ + \angle 1 = 180^\circ$

Simplifying:

$124^\circ + \angle 1 = 180^\circ$

Solving for $\angle 1$:

$\angle 1 = 180^\circ - 124^\circ = 56^\circ$

Thus, $\angle 1 = 56^\circ$, which corresponds to option A.

Would you like further details or have any other questions?

Here are 5 related questions to consider:

1. What are the rules for the sum of interior angles in a polygon?
2. How do you calculate the missing angle in any triangle if two angles are known?
3. Can the sum of angles in a triangle ever be more or less than 180°?
4. How does the concept of supplementary angles apply in this scenario?
5. How can you use exterior angles to find unknown interior angles?

Tip: In any triangle, knowing two angles immediately allows you to find the third by subtracting their sum from 180°.