To calculate the measure of the marked angle in the triangle:

1. Sum of angles in a triangle: The sum of all interior angles of a triangle is always $180^\circ$.

2. The triangle provided has one known angle, $69.5^\circ$, and the marked angle ($?^\circ$) at the apex, as well as two identical angles at the base.

3. Identical base angles: The two base angles are equal, as indicated by the symmetry of the triangle and markings (isosceles triangle).

Let:

• The marked angle (apex) = $x$,
• Each base angle = $y$.

Using the sum of angles: $x + 2y = 180^\circ$

From the diagram, one base angle is labeled $69.5^\circ$, so: $2(69.5) + x = 180$

$x = 180 - 139$

$x = 41^\circ$

Answer:

The marked angle is $41^\circ$.

Let me know if you want further explanation! Here are 5 related questions to explore:

1. How can you generalize solving for angles in isosceles triangles?
2. What are the properties of equilateral triangles compared to isosceles triangles?
3. How can you calculate exterior angles of triangles?
4. How does the triangle angle sum theorem apply in non-Euclidean geometry?
5. How can angles be used to determine triangle similarity?

Tip: Always check the type of triangle (isosceles, equilateral, scalene) to simplify angle calculations.