🔺 Polygons with Angles x° and (360–x)° Only

Concept:

In a polygon, each interior angle must be less than 360°. If a polygon has two types of angles: \[ x^\circ \quad \text{and} \quad (360^\circ - x^\circ) \] then the sum of all interior angles is still governed by the formula: \[ \text{Sum} = (n - 2) \times 180^\circ \] If there are \( a \) angles of measure \( x^\circ \), and \( b \) angles of measure \( 360 - x^\circ \), the total becomes: \[ a \cdot x^\circ + b \cdot (360^\circ - x^\circ) = (a + b - 2) \cdot 180^\circ \]

🧩 Example-Based Questions

Q1. If a polygon has x = 90° and y = 270°, then prove that x – y = 4.

Explanation:

Total number of angles = x + y
Sum of angles = \(90x + 270y\)
Equating with sum of polygon: \((x + y - 2) \cdot 180\)

\[ 90x + 270y = (x + y - 2) \cdot 180 \Rightarrow x + 3y = 2x + 2y - 4 \Rightarrow x - y = 4 \]

Q2. In a polygon having 60° and 300° as interior angles, show that x – y = 3.

Same approach:
Let there be \( x \) angles of 60° and \( y \) angles of 300°. \[ 60x + 300y = (x + y - 2) \cdot 180 \Rightarrow x + 5y = 3x + 3y - 6 \Rightarrow x - y = 3 \]

Q3. If a polygon has 120° and 240° as interior angles, show that x – y = 6.

Let counts of 120° and 240° be \( x \) and \( y \) respectively: \[ 120x + 240y = (x + y - 2) \cdot 180 \Rightarrow x - y = 6 \]

🧠 Generalized Formula

If a polygon has two types of angles: \( x^\circ \) and \( (360 - x)^\circ \), then:

\[ x - y = \frac{360}{180 - x} \] This gives a direct relation between how many of each angle must be present.

🎯 Application-Based Problem

Q4. In a decagon (10 sides), it consists of only 90° and 270° interior angles. How many 90° angles are there?

We are given \( x + y = 10 \) and from earlier: \( x - y = 4 \)

Solving:

\[ x + y = 10, \quad x - y = 4 \Rightarrow 2x = 14 \Rightarrow x = 7 \]

✅ Answer: There are 7 angles of 90° and 3 angles of 270°.

📌 Final Thoughts

Problems involving angles \( x^\circ \) and \( 360^\circ - x^\circ \) are tricky but logical. These concepts are valuable in CAT, SSC, and other aptitude exams where geometry meets logical reasoning. The general formula lets you derive relations quickly.

Practice pattern-based questions like these to improve speed and accuracy!

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