Amiya Kumar 
Understanding Polygon Angles and Arithmetic Progression in Geometry
🔹 Sum of all exterior angles of any polygon is always:
\[ \text{Sum of Exterior Angles} = 360^\circ \]
🔹 Sum of all interior angles of an n-sided polygon:
\[ \text{Sum of Interior Angles} = (n - 2) \times 180^\circ \]
Angles in A.P. – A Beautiful Pattern
When angles of a polygon are in Arithmetic Progression (A.P.), we can apply the same formulas with a twist:
- If 3 angles of triangle are in A.P., say \( a - d, a, a + d \), then:
- \[ (a - d) + a + (a + d) = 3a = 180^\circ \Rightarrow a = 60^\circ \]
Practice Question on Pentagon with Angles in A.P.
📝 Question:
The five interior angles of a pentagon are in Arithmetic Progression. If the smallest angle is 78°, find the other four angles.
📌 Step 1: Use angle sum formula
\[ \text{Sum of interior angles of a pentagon} = (5 - 2) \times 180 = 540^\circ \]
📌 Step 2: Let the five angles be:
\[ a - 2d,\ a - d,\ a,\ a + d,\ a + 2d \]
Given: \( a - 2d = 78^\circ \)
📌 Step 3: Use the total sum
\[ \text{Sum} = (a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 5a = 540^\circ \Rightarrow a = 108^\circ \] Now substitute back: \[ a - 2d = 78 \Rightarrow 108 - 2d = 78 \Rightarrow d = 15 \] So the five angles are: \[ 78^\circ,\ 93^\circ,\ 108^\circ,\ 123^\circ,\ 138^\circ \]
Bonus Tips:
- Exterior angle = 180° – interior angle
- If polygon has equal angles (regular polygon), each interior angle = \(\frac{(n-2)\times180}{n}\)
Conclusion
Understanding the properties of interior and exterior angles, especially when angles are in A.P., opens up a world of interesting problems in competitive exams like CAT, SSC, and banking. Practice identifying these patterns and apply the sum formulas strategically!
Keep learning, and always look for patterns!
