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We are given an irregular pentagon with various angles labeled. The goal is to calculate the angle \( m \). Let's break this down step by step using the known angles:

1. **Known angles**:
   - \( 86^\circ \)
   - \( 72^\circ \)
   - \( 25^\circ \)

2. **Internal angle of a polygon**:
   - The sum of the internal angles of a pentagon is given by the formula:
     \[
     \text{Sum of internal angles} = (n-2) \times 180^\circ
     \]
     For a pentagon \( n = 5 \), so:
     \[
     \text{Sum of internal angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ
     \]

3. **Four of the five angles are given**:
   Adding up the angles we know:
   \[
   25^\circ + 86^\circ + 72^\circ = 183^\circ
   \]

4. **Remaining internal angles**:
   We know the interior angle at the bottom-left corner is \( 90^\circ \) (since it's a right angle), so the total for the four known angles is:
   \[
   183^\circ + 90^\circ = 273^\circ
   \]

5. **Calculate \( m \)**:
   Using the total internal angles formula, we know the sum must be \( 540^\circ \). The remaining angle \( m \) can be found by subtracting the sum of the four known angles:
   \[
   m = 540^\circ - 273^\circ = 267^\circ
   \]

Thus, the angle \( m \) is \( \boxed{267^\circ} \).

Do you need further clarification on any steps? 

Here are some related questions:
1. How can the internal angle sum formula be applied to other polygons?
2. What would happen if the pentagon were regular instead of irregular?
3. How do external angles relate to internal angles in polygons?
4. Why does the sum of angles always hold for polygons, and how is it derived?
5. How can we extend this to more complex polygons?

**Tip**: Always double-check angle sums when dealing with polygons to avoid small calculation errors!

An irregular pentagon is shown. Calculate the size of the angle marked m.

Learn how to calculate the missing angle m in an irregular pentagon using the sum of interior angles formula. This problem involves known angles of 25°, 86°, and 72°, and uses basic geometry principles to find the solution. Suitable for students in Grades 8-10.

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