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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!

Find the measure of angle x in the figure below: The figure shows two triangles forming an 'hourglass' shape. There are angles of 60° and 50° at the base of the lower triangle.

Learn how to find the measure of angle x in a geometry problem involving triangles and vertical angles. Use the Triangle Sum Theorem and Vertical Angles Theorem to solve the problem. Suitable for students in Grades 6-8.

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