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The given problem involves an isosceles triangle, \( \triangle OPQ \), where \( \overline{OP} \cong \overline{QO} \) and \( m\angle Q = 12^\circ \). We are asked to find \( m\angle O \).

### Step-by-step Solution:
1. Since \( \overline{OP} \cong \overline{QO} \), triangle \( \triangle OPQ \) is an **isosceles triangle**. This means that the angles opposite the equal sides are equal.
   - Therefore, \( \angle P = \angle O \).

2. The sum of angles in any triangle is always \( 180^\circ \).
   - \( m\angle O + m\angle P + m\angle Q = 180^\circ \).

3. We know that \( m\angle Q = 12^\circ \), and \( m\angle O = m\angle P \) since it's an isosceles triangle. Let \( m\angle O = m\angle P = x \).

4. Now we can set up the equation:
   \[
   x + x + 12^\circ = 180^\circ
   \]
   \[
   2x + 12^\circ = 180^\circ
   \]
   \[
   2x = 180^\circ - 12^\circ
   \]
   \[
   2x = 168^\circ
   \]
   \[
   x = \frac{168^\circ}{2} = 84^\circ
   \]

Thus, \( m\angle O = 84^\circ \).

Would you like any further explanation or clarification?

Here are 5 related questions to explore the topic further:
1. What are the properties of an isosceles triangle?
2. How do we solve for angles in non-isosceles triangles?
3. Can a triangle have more than one obtuse angle?
4. How do we use the exterior angle theorem in triangles?
5. How does the sum of interior angles change for polygons with more than three sides?

**Tip**: Always remember that the sum of the interior angles in any triangle is \(180^\circ\).

In triangle OPQ, OP ≅ QO and m∠Q = 12°. Find m∠O.

This problem explores the solution of finding m∠O in isosceles triangle OPQ, where OP ≅ QO and m∠Q = 12°. Using the Triangle Angle Sum Theorem and properties of isosceles triangles, the problem is solved step by step.

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