The Triangles and its Properties Ex 6.3 - CBSE Class 7 Maths

Welcome to YoLearn AI! In Chapter 6, Exercise 6.3 of Class 7 Maths, we explore one of the most fundamental properties of geometry: the Angle Sum Property of a Triangle. Have you ever wondered why every triangle, no matter its size or shape—whether it is tiny, huge, acute-angled, or obtuse-angled—always has angles that sum up to exactly 180 degrees? This exercise focuses on mastering this brilliant rule to find unknown interior angles of various triangles. By understanding this concept, you will unlock the ability to solve algebraic equations involving angles and build a strong foundation for high school geometry. In this guide, our YoLearn AI Tutor will walk you through step-by-step proofs, clear visual concepts, and worked-out solutions so you can confidently tackle any problem in your CBSE exams. Let's dive in!

Understanding the Angle Sum Property of a Triangle

The Angle Sum Property states that the sum of the measures of the three interior angles of a triangle is always equal to 180 degrees. Mathematically, if we have a triangle ABC with angles A, B, and C, then Angle A + Angle B + Angle C = 180 degrees. To prove this logically, imagine drawing a straight line parallel to the base BC passing through the top vertex A. By using the properties of alternate interior angles formed by a transversal intersecting parallel lines, we can easily show that the three interior angles lie perfectly on a straight line at vertex A, summing up to 180 degrees. This property helps us find any unknown angle when the other two are given. It is a vital tool used across civil engineering, architecture, computer graphics, and physics.

Step-by-Step Method to Solve Exercise 6.3 Problems

  1. Identify the Given Values — Locate and write down the measures of the known angles given in the triangle diagram.
  2. Formulate the Equation — Set up your equation using the Angle Sum Property: x + Angle 1 + Angle 2 = 180 degrees, where x represents the unknown angle.
  3. Simplify the Constants — Add the values of the two known angles together to simplify the numerical part of your equation.
  4. Solve for the Unknown Variable — Subtract the sum of the known angles from 180 degrees to find the value of x. Always write the final answer with the degree symbol.

Worked Examples from Ex 6.3 Concepts

  • Example 1: Find the value of the unknown x in a triangle where two angles are 50 degrees and 60 degrees. Step-by-step Solution: 1. Let the angles of the triangle be 50°, 60°, and x. 2. By the Angle Sum Property: 50° + 60° + x = 180°. 3. Combine the terms: 110° + x = 180°. 4. Isolate x: x = 180° - 110° = 70°. Thus, the unknown angle is 70°.
  • Example 2: In a right-angled triangle, one acute angle is 30 degrees. Find the other acute angle. Step-by-step Solution: 1. A right-angled triangle always contains one 90° angle. 2. Let the unknown angle be x. The other angles are 90° and 30°. 3. Apply the property: x + 90° + 30° = 180°. 4. Simplify: x + 120° = 180°. 5. Solve: x = 180° - 120° = 60°. Thus, the other acute angle is 60°.

Crucial Exam Tips & Common Pitfalls

  1. Don't Forget the Right Angle Sign: In CBSE question papers, a square corner symbol at a vertex represents a 90-degree angle. Don't leave it out of your calculations just because the number '90' is not explicitly written!
  2. Double Check with Simple Addition: Once you calculate your value of x, always add all three final angles up to verify if they equal exactly 180 degrees. If they add up to 170 degrees or 190 degrees, you made an arithmetic error.
  3. Isosceles Triangles: Remember that angles opposite to equal sides are also equal. If a triangle is marked with two equal sides, it means their opposite angles are also equal (often both labeled as x).

Practice Questions with Solutions

  • Q: Find the value of x in a triangle with interior angles 2x, x, and 90 degrees. A: Step 1: Write down the Angle Sum Property equation: 2x + x + 90 = 180. Step 2: Combine like terms (the algebraic terms): 3x + 90 = 180. Step 3: Subtract 90 from both sides: 3x = 180 - 90 = 90. Step 4: Divide by 3 to solve for x: x = 90 / 3 = 30. Final answer: x = 30°
  • Q: An equilateral triangle has three equal angles, each labeled as x. Find the value of x. A: Step 1: Set up the equation using the sum of the three equal angles: x + x + x = 180. Step 2: Combine the variables: 3x = 180. Step 3: Solve for x by dividing both sides by 3: x = 180 / 3 = 60. Final answer: x = 60°
  • Q: One of the angles of a triangle is 80 degrees and the other two angles are equal. Find the measure of each of the equal angles. A: Step 1: Let the two equal angles be x and x. Step 2: Use the Angle Sum Property: x + x + 80 = 180. Step 3: Simplify the equation: 2x + 80 = 180. Step 4: Subtract 80 from both sides: 2x = 100. Step 5: Solve for x: x = 100 / 2 = 50. Final answer: Each of the equal angles measures 50°
  • Q: In a triangle, the angles are in the ratio 1:2:3. Find all three angles. A: Step 1: Let the three angles be 1x, 2x, and 3x. Step 2: Apply the Angle Sum Property: 1x + 2x + 3x = 180. Step 3: Add the terms together: 6x = 180. Step 4: Solve for x: x = 180 / 6 = 30. Step 5: Find the individual angles: First angle = 1x = 30°, Second angle = 2x = 60°, Third angle = 3x = 90°. Final answer: The angles are 30°, 60°, and 90°

Frequently Asked Questions

What is the Angle Sum Property of a triangle?

The Angle Sum Property states that the sum of the three interior angles of any triangle is always equal to 180 degrees. This rule applies to all triangles regardless of their size, orientation, or shape.

Can a triangle have two right angles?

No, a triangle cannot have two right angles. Since the sum of all three angles must be exactly 180 degrees, having two 90-degree angles would already sum up to 180 degrees, leaving 0 degrees for the third angle, which is geometrically impossible.

How is the exterior angle property related to the angle sum property?

The exterior angle of a triangle equals the sum of its two interior opposite angles. Both properties are structurally linked to the linear pair axiom and parallel line postulates in geometry.