The given figure shows two triangles forming an "hourglass" shape. You're asked to find the measure of angle $x$, with two known angles at the bottom: $60^\circ$ and $50^\circ$. Here's how we solve it:

1. Step 1: Use the Triangle Sum Theorem. In any triangle, the sum of the angles is always $180^\circ$. For the lower triangle, let's call the third angle at the top of this triangle $y$.

   The sum of the angles in the lower triangle is: $y + 60^\circ + 50^\circ = 180^\circ$ Solving for $y$: $y = 180^\circ - 60^\circ - 50^\circ = 70^\circ$

2. Step 2: Use Vertical Angles. Vertical angles are congruent, meaning they have the same measure. Angle $x$ and angle $y$ are vertical angles, so: $x = y = 70^\circ$

Thus, the measure of angle $x$ is $70^\circ$.

Would you like further explanation on any of the steps or concepts involved?

Here are 5 related questions for further exploration:

1. What is the Triangle Sum Theorem and how is it applied?
2. What are vertical angles and why are they always equal?
3. How do you calculate angles in non-triangle shapes?
4. If one of the angles in the lower triangle was changed, how would that affect $x$?
5. Can the concept of complementary and supplementary angles be applied in this problem?

Tip: Remember that vertical angles are always equal when two lines intersect, which is a helpful rule in geometry!