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We are given a triangle, where \(PQ\) is the perpendicular bisector of \(AC\), and we are tasked with finding the measure of \(\angle C\).

### Given:
- \(\angle A = 40^\circ\)
- \(\angle B = 80^\circ\)
- \(PQ\) is the perpendicular bisector of \(AC\), which implies \(\triangle APC\) is isosceles, meaning \(\angle APC = \angle A\).

### Steps:
1. **Find the measure of \(\angle P\):**
   Since \(\triangle APB\) is a triangle, we can use the sum of angles in a triangle, which is \(180^\circ\):
   \[
   \angle A + \angle B + \angle P = 180^\circ
   \]
   Substituting the known values:
   \[
   40^\circ + 80^\circ + \angle P = 180^\circ
   \]
   Simplifying:
   \[
   120^\circ + \angle P = 180^\circ
   \]
   \[
   \angle P = 60^\circ
   \]

2. **Use the properties of isosceles triangles:**
   Since \(PQ\) is the perpendicular bisector of \(AC\), triangle \(\triangle APC\) is isosceles. Therefore, \(\angle A = \angle C\).

   So, \(\angle C = 40^\circ\).

Thus, the measure of \(\angle C\) is **40°**.

Would you like more details on any step?

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Here are five related questions for practice:
1. What would be the measure of \(\angle C\) if \(\angle A\) was 50° instead?
2. How can we verify that \(PQ\) is the perpendicular bisector of \(AC\) using geometric properties?
3. If \(\angle B\) was increased to 90°, how would that affect \(\triangle APC\)?
4. What other methods could you use to confirm that \(\triangle APC\) is isosceles?
5. How would the solution change if \(PQ\) were not a perpendicular bisector?

**Tip**: Always check for isosceles triangles when you know a line bisects a side and is perpendicular. This gives you equal angles to use in your calculations!

En la siguiente figura, PQ es mediatriz de AC. Halla m∠C.

Learn how to find the measure of angle C in a triangle where PQ is the perpendicular bisector of AC. This problem involves key geometric concepts like the sum of angles in a triangle and the properties of isosceles triangles. Suitable for students in Grades 8-10.

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