Understanding Polygon Angles and Arithmetic Progression in Geometry

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.

📌 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!