NCERT Solutions Class 8 Maths Understanding Quadrilaterals Exercise 3.2

Welcome back, maths champions! In this guide, we dive deep into NCERT Class 8 Maths Chapter 3, Exercise 3.2. This exercise focuses on a fascinating property of polygons: the sum of their exterior angles. While the sum of interior angles changes as you add more sides to a polygon, the sum of the exterior angles is remarkably constant. No matter if your polygon has 3 sides (a triangle), 4 sides (a quadrilateral), or 100 sides (a hectagon), the sum of its exterior angles is always exactly 360 degrees. Mastering this powerful rule will help you solve geometry problems in seconds. In this tutorial, the YoLearn AI Tutor will walk you through the core concepts, prove the exterior angle property, show you step-by-step solutions for exercise problems, and give you practice questions to ace your school exams.

The Magic of Exterior Angles: The 360° Rule

An exterior angle is formed when we extend one of the sides of a polygon outwards. If you start at one vertex, walk along the perimeter, and make a turn at each vertex to keep going along the boundary, you will eventually end up pointing in the exact same direction you started. This means you have made one complete turn, which is exactly 360 degrees! Therefore, the sum of the measures of the exterior angles of any convex polygon is always 360°. For a regular polygon with 'n' sides, all sides and interior angles are equal. Consequently, all its exterior angles are also equal. This gives us two powerful formulas:

  1. Measure of each exterior angle = 360° / n
  2. Number of sides (n) = 360° / Measure of each exterior angle.

How to Solve Problems on Exterior Angles

  1. Identify the Given Value — Read the problem carefully to find if you are given the number of sides (n), the value of each exterior angle, or a set of unequal exterior angles with one unknown variable 'x'.
  2. Apply the Correct Formula — If finding the number of sides, use: n = 360° / (each exterior angle). If finding the measure of each exterior angle, use: Angle = 360° / n. If given a diagram, sum all given exterior angles up to 360°.
  3. Convert Interior Angles if Necessary — Remember that an interior angle and its corresponding exterior angle form a straight line (linear pair). Thus: Exterior Angle + Interior Angle = 180°.
  4. Perform the Calculation — Solve the algebraic equation or divide 360 by the given number to find the final result.

Important Exam Tips & Common Mistakes

  1. Don't Confuse Interior and Exterior Sums: Remember that the sum of interior angles varies with the number of sides, calculated as (n - 2) 180°. However, the sum of exterior angles is always* 360° regardless of the number of sides.
  2. Linear Pair Check: At any vertex, the interior and exterior angle must add up to 180°. If your calculated interior and exterior angles don't sum to 180°, double-check your steps.
  3. Regular Polygons Only: The formula (360° / n) works only for regular polygons. If the polygon is irregular, you must add up the individual angles and set the sum equal to 360°.

Practice Questions with Solutions

  • Q: Find the value of x in a polygon where the exterior angles are 125°, 125°, and x°. A: Step 1: Identify the type of polygon. Since there are 3 exterior angles, this is a triangle. Step 2: Recall that the sum of all exterior angles of any polygon is 360°. Step 3: Set up the equation: 125° + 125° + x° = 360°. Step 4: Simplify the equation: 250° + x° = 360°. Step 5: Subtract 250° from both sides to find x: x = 360° - 250° = 110°. Final answer: x = 110°.
  • Q: Find the number of sides of a regular polygon if each of its exterior angles is 45°. A: Step 1: Note the given exterior angle value, which is 45°. Step 2: Recall the formula for the number of sides of a regular polygon: n = 360° / (each exterior angle). Step 3: Substitute the value into the formula: n = 360 / 45. Step 4: Perform the division: n = 8. Final answer: The regular polygon has 8 sides (it is an octagon).
  • Q: Is it possible to have a regular polygon with a measure of each exterior angle as 22°? Give reasons. A: Step 1: Use the formula for the number of sides: n = 360° / (each exterior angle). Step 2: Substitute 22° into the formula: n = 360 / 22. Step 3: Calculate the value: n = 16.36. Step 4: Since the number of sides (n) of a polygon must be a whole number, 22 is not a divisor of 360. Final answer: No, it is not possible to have a regular polygon with an exterior angle of 22° because 360 is not perfectly divisible by 22.
  • Q: Find the measure of each interior angle of a regular polygon with 15 sides. A: Step 1: First, find the measure of each exterior angle using the formula: Exterior Angle = 360° / n. Step 2: Substitute n = 15: Exterior Angle = 360° / 15 = 24°. Step 3: Use the linear pair relationship to find the interior angle: Interior Angle = 180° - Exterior Angle. Step 4: Calculate: Interior Angle = 180° - 24° = 156°. Final answer: The measure of each interior angle is 156°.

Frequently Asked Questions

What is the sum of the exterior angles of any polygon?

The sum of the exterior angles of any convex polygon is always 360 degrees, regardless of how many sides the polygon has.

How do you find the exterior angle if the interior angle is given?

An interior angle and its adjacent exterior angle form a straight line, meaning they are supplementary. You can find the exterior angle by subtracting the interior angle from 180 degrees.

Does the 360° rule apply to irregular polygons as well?

Yes, the sum of all exterior angles is 360 degrees for both regular and irregular convex polygons. However, only regular polygons have equal individual exterior angles.