Maths activity class 9. How to verify that the sum of the angle of a triangle is 180°

To verify that the sum of the angles of a triangle is 180°, you can use a simple geometric method involving parallel lines. Here's a step-by-step activity you can conduct: ### Materials Needed: - A triangle (can be drawn on paper or made with a ruler and compass) - A ruler - A protractor - A piece of paper ### Steps: 1. **Draw a Triangle**: Draw a triangle, label its vertices as A, B, and C. 2. **Label the Angles**: Label the angles at the vertices as ∠A, ∠B, and ∠C. 3. **Extend a Side**: Extend one side of the triangle (for example, side BC) to form a line. 4. **Draw a Parallel Line**: Use a ruler to draw a line through vertex A that is parallel to line BC. 5. **Label Angles**: Let the new line intersect the extension of line BC at point D. Now, we will label some angles: - ∠CAB (which is ∠A) - ∠CBA (which is ∠B) - The angle formed where line AD intersects the extended line BC, which is equal to ∠A because they are corresponding angles. - The angle formed at vertex B with line AD is equal to ∠C because they are also corresponding angles. 6. **Identify Relationships**: According to the properties of parallel lines: - The sum of ∠A, ∠B, and ∠C will add up to form a straight angle (180°) because they are on a straight line. 7. **Measure Angles**: If you have drawn the triangle accurately, you can measure the angles ∠A, ∠B, and ∠C using a protractor to confirm that their sum equals 180°: \[ \text{Sum of angles} = \angle A + \angle B + \angle C = 180° \] 8. **Conclusion**: Conclude that the sum of the angles in any triangle is 180°, as demonstrated by the properties of parallel lines and the corresponding angles. ### Additional Verification (Using the Exterior Angle Theorem): You can also demonstrate this using the Exterior Angle Theorem: 1. **Draw an Exterior Angle**: Extend one side of the triangle (like BC) to point D. 2. **Label Angles**: The exterior angle ∠ADC is equal to the sum of the two opposite interior angles ∠A and ∠B: \[ \angle ADC = \angle A + \angle B \] 3. **Since angle ADC is also equal to 180° (because it's a straight line)**, you can conclude: \[ \angle A + \angle B + \angle C = 180° \] Using either of these methods, you can effectively verify that the sum of the angles of a triangle is always 180°.