NCERT Class 7 Maths: The Triangles and its Properties Ex 6.2

Hello young mathematicians! In Class 7 Maths Chapter 6, "The Triangles and its Properties," you're about to uncover some fascinating secrets about triangles. Specifically, Exercise 6.2 focuses on a very important concept called the Exterior Angle Property of a Triangle. Understanding this property is like gaining a superpower for solving problems involving angles in triangles!

Why is this important? Triangles are everywhere, from the pyramids to the slices of pizza you eat! Knowing how their angles behave helps us understand shapes better and lays the foundation for more advanced geometry. By the end of this page, you will master what an exterior angle is, how it relates to interior angles, and confidently solve problems from NCERT Exercise 6.2. Let's dive in and make geometry fun and easy!

What is the Exterior Angle Property of a Triangle?

Imagine a triangle, let's call it ABC. If you extend any one side of this triangle, you create an angle outside the triangle. This is what we call an exterior angle. For example, if you extend side BC to a point D, then the angle ACD is an exterior angle.

Now, inside the triangle, there are three angles: ∠A, ∠B, and ∠C. The angle ∠ACB is right next to our exterior angle ∠ACD. These two angles, ∠ACB and ∠ACD, are adjacent (next to each other) and form a linear pair, so their sum is 180 degrees.

The magic happens when we look at the other two angles inside the triangle, ∠A and ∠B. These are called the interior opposite angles to the exterior angle ∠ACD. The Exterior Angle Property states that:

An exterior angle of a triangle is equal to the sum of its two interior opposite angles.

So, for our example, ∠ACD = ∠A + ∠B. This property is incredibly useful for finding unknown angles in various geometric figures without needing to know all the angles inside the triangle first. It simplifies many calculations!

Steps to Apply the Exterior Angle Property

  1. Step 1: Identify the Exterior Angle — Look for the angle formed outside the triangle by extending one of its sides. This is your exterior angle. There will be an adjacent interior angle next to it.
  2. Step 2: Identify the Two Interior Opposite Angles — Once you've found the exterior angle and its adjacent interior angle, the other two angles inside the triangle are the interior opposite angles. They are not next to the exterior angle.
  3. Step 3: Formulate the Equation — According to the property, the exterior angle is equal to the sum of the two interior opposite angles. Write this down as an equation, often with an unknown variable like 'x'.
  4. Step 4: Solve for the Unknown — Use simple algebra to solve the equation and find the value of the unknown angle. Remember your basic addition and subtraction rules.

Worked Examples: Using the Exterior Angle Property

  • Example 1: In triangle ABC, side BC is extended to D. If ∠A = 50°, ∠B = 60°, find the exterior angle ∠ACD. Given: Interior opposite angles are ∠A = 50° and ∠B = 60°. Property: Exterior Angle ∠ACD = Sum of interior opposite angles. Calculation: ∠ACD = ∠A + ∠B = 50° + 60° = 110°. Answer: The exterior angle ∠ACD is 110°.
  • Example 2: In triangle PQR, side QR is extended to S. The exterior angle ∠PRS is 120°. If ∠P = 70°, find ∠Q. Given: Exterior angle ∠PRS = 120°, Interior opposite angle ∠P = 70°. Property: Exterior Angle ∠PRS = ∠P + ∠Q. Formulate Equation: 120° = 70° + ∠Q. Solve for ∠Q: ∠Q = 120° - 70° = 50°. * Answer: The angle ∠Q is 50°.
  • Example 3: Find the value of x in the following figure: A triangle has interior angles 35° and 85°. The exterior angle corresponding to the third interior angle is marked as x. Given: Interior opposite angles are 35° and 85°. Property: Exterior angle x = Sum of interior opposite angles. Calculation: x = 35° + 85° = 120°. Answer: The value of x is 120°.

Exam Tip: Avoiding Common Mistakes

One of the most frequent mistakes students make is confusing the interior adjacent angle with the interior opposite angles. Remember, the exterior angle property only uses the two interior angles that are not next to the exterior angle. Always clearly identify your exterior angle and then look across the triangle for the two angles that are "opposite" to it. Also, double-check your addition and subtraction. A small calculation error can lead to a wrong final answer, even if your concept is clear!

Practice Questions with Solutions

  • Q: In a triangle XYZ, side YZ is extended to point W. If ∠X = 45° and ∠Y = 75°, what is the measure of the exterior angle ∠XZW? A: Step 1: Identify the exterior angle (∠XZW) and the interior opposite angles (∠X and ∠Y). Step 2: Apply the Exterior Angle Property: ∠XZW = ∠X + ∠Y. Step 3: Substitute the given values: ∠XZW = 45° + 75°. Step 4: Calculate the sum: ∠XZW = 120°. Final answer: The exterior angle ∠XZW is 120°.
  • Q: The exterior angle of a triangle at one vertex is 115°. One of its interior opposite angles is 60°. Find the other interior opposite angle. A: Step 1: Let the exterior angle be E = 115° and one interior opposite angle be A = 60°. Step 2: Let the other interior opposite angle be B. The Exterior Angle Property states E = A + B. Step 3: Substitute the values into the equation: 115° = 60° + B. Step 4: Solve for B: B = 115° - 60° = 55°. Final answer: The other interior opposite angle is 55°.
  • Q: In triangle LMN, side MN is produced to P. If ∠L = 90° and ∠MLP (exterior angle) = 140°, what is the measure of ∠M? A: Step 1: The exterior angle is ∠MLP = 140°. The interior opposite angles are ∠M and ∠L. Step 2: Apply the Exterior Angle Property: ∠MLP = ∠M + ∠L. Step 3: Substitute known values: 140° = ∠M + 90°. Step 4: Solve for ∠M: ∠M = 140° - 90° = 50°. Final answer: The measure of ∠M is 50°.
  • Q: A triangle has interior angles 30° and 70°. What is the measure of the exterior angle at the third vertex? A: Step 1: The two given angles (30° and 70°) are the interior opposite angles to the exterior angle we need to find. Step 2: Apply the Exterior Angle Property: Exterior Angle = 30° + 70°. Step 3: Calculate the sum: Exterior Angle = 100°. Final answer: The exterior angle at the third vertex is 100°.

Frequently Asked Questions

What is an exterior angle of a triangle?

An exterior angle is formed when one side of a triangle is extended. It is the angle outside the triangle and forms a linear pair with the adjacent interior angle.

What is the Exterior Angle Property?

This property states that an exterior angle of a triangle is always equal to the sum of its two interior opposite angles. It's a fundamental concept for understanding triangle geometry.

How can I identify the interior opposite angles?

To find the interior opposite angles, first locate the exterior angle and its adjacent interior angle. The other two interior angles of the triangle, which are not adjacent to the exterior angle, are the interior opposite angles.

Does this property work for all types of triangles?

Yes, the Exterior Angle Property applies to all types of triangles, including equilateral, isosceles, and scalene triangles. It's a universal rule in Euclidean geometry.