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en la siguiente figura PQ es mediatriz de AC. Hallar medida del angulo c

In this triangle, we are tasked with finding the measure of angle \( m \angle C \). Here's a step-by-step analysis:

1. **Given Data**:
   - \( PQ \) is the perpendicular bisector (mediatriz) of \( AC \), meaning \( \angle PQA = \angle PQC = 90^\circ \).
   - \( \angle A = 40^\circ \) and \( \angle B = 80^\circ \) are given.

2. **Key Observations**:
   - Since \( PQ \) is the perpendicular bisector, \( AP = PC \), and \( \angle PQA = 90^\circ \), so triangle \( AQP \) is a right triangle.
   - Similarly, triangle \( PQC \) is also a right triangle with \( \angle PQC = 90^\circ \).

3. **Using Triangle Sum Property**:
   Let's first focus on triangle \( ABP \):
   - The sum of the angles in any triangle is \( 180^\circ \). Therefore:
     \[
     \angle PAB + \angle ABP + \angle BPA = 180^\circ
     \]
   - Plugging in the known values:
     \[
     40^\circ + 80^\circ + \angle BPA = 180^\circ
     \]
     Solving for \( \angle BPA \):
     \[
     \angle BPA = 180^\circ - 40^\circ - 80^\circ = 60^\circ
     \]

4. **Finding \( \angle C \)**:
   Since \( \triangle APC \) is isosceles (because \( AP = PC \)), \( \angle APC = \angle CPA \). We know from triangle sum property:
   \[
   \angle PAC + \angle APC + \angle CPA = 180^\circ
   \]
   Let \( \angle APC = \angle CPA = x \). Thus:
   \[
   40^\circ + 2x = 180^\circ
   \]
   Solving for \( x \):
   \[
   2x = 140^\circ \quad \Rightarrow \quad x = 70^\circ
   \]
   Therefore, \( \angle APC = 70^\circ \).

Thus, the measure of \( m \angle C = 70^\circ \).

Let me know if you'd like further clarification on any step! Here are some related questions to explore:

1. How do you identify isosceles triangles using a perpendicular bisector?
2. What is the significance of the triangle sum property?
3. Can the concept of symmetry help in finding unknown angles in any triangle?
4. What are other ways to confirm that \( \angle C \) is 70°?
5. How would the problem change if \( PQ \) weren't a perpendicular bisector?

**Tip**: Always verify if special lines like perpendicular bisectors create isosceles triangles—they simplify calculations greatly!

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

Learn how to find the angle m∠C in a triangle where PQ is the perpendicular bisector of AC. This geometry problem involves key concepts such as the triangle sum theorem and properties of isosceles triangles. Suitable for students in Grades 7-9.

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