Maths class 9 - Activity 5: To verify that the sum of the angle of a triangle is 180°. Using paper cutting and pasting method. Including - Materials needed, method of construction, demonstration, observation and the application

### Activity 5: Verifying that the Sum of the Angles of a Triangle is 180° #### Materials Needed: 1. A sheet of paper (preferably A4 size) 2. A ruler 3. A pair of scissors 4. A protractor 5. A pencil 6. Glue or tape 7. A compass (optional, for drawing perfect circles) #### Method of Construction: 1. **Draw a Triangle:** - Use a ruler and a pencil to draw any triangle (for example, ABC) on the paper. Ensure it is not a right-angled triangle for more general applicability. 2. **Label the Angles:** - Label the vertices of the triangle as A, B, and C, and label the angles at those vertices as ∠A, ∠B, and ∠C respectively. 3. **Cut out the Triangle:** - Carefully cut along the edges of the triangle to create a paper triangle. 4. **Cut the Angles:** - Starting from vertex A, cut along the segment that bisects angle A so that you have two pieces: one part for angle A and the other part for the remaining angles (B and C). - Repeat this for angles B and C. You should now have three angle pieces from your triangle. 5. **Position the Angles:** - Arrange the three angle pieces (angle A, angle B, and angle C) on a flat surface such that their pointed ends meet at a single point. Try to align them as if they are adjacent to each other. 6. **Paste the Angles:** - Use glue or tape to fix the angles at the meeting point to keep them in place. #### Demonstration: - **Visual Explanation:** - Move the three angles together, and demonstrate that they create a straight line (180°). You can use a protractor to measure the angles of the triangle you constructed and confirm that when added, they sum up to 180°. #### Observation: - **What You’ll Notice:** - As you arrange the three angle pieces, they will fit perfectly together to form a straight line. You should observe that ∠A + ∠B + ∠C = 180°. This is a practical verification that the sum of the angles in a triangle indeed equals 180°. #### Application: - **Understanding the Concept:** - This activity helps solidify the concept of angle sums and how geometric properties are interconnected. It’s applicable not just in triangles but also serves as a foundation for more complex geometric proofs and concepts in higher mathematics. Moreover, this understanding is critical in various fields such as architecture, engineering, and various branches of physics. By engaging in this hands-on activity, learners will have a tangible experience that reinforces their understanding of fundamental geometric principles, while also developing skills in measurement, cutting, and arranging shapes. Update (2024-11-19): ### Activity 5: Verifying that the Sum of the Angles of a Triangle is 180° #### Materials Needed: - A sheet of paper - Scissors - A ruler - A protractor - A pencil - A glue stick or tape #### Method of Construction: 1. **Draw a Triangle:** - Use the ruler to draw a triangle on the sheet of paper. Label the vertices of the triangle as A, B, and C. For instance, let the triangle be an arbitrary triangle with vertices A(0,0), B(4,0), and C(2,3). 2. **Label Angles:** - Use the protractor to measure and label the angles at each vertex of the triangle. Label the angles as ∠A, ∠B, and ∠C. 3. **Cut Out the Triangle:** - Carefully cut out the triangle along the edges using scissors. This creates a triangular piece of paper. 4. **Cut and Paste Method:** - Cut the triangle into three parts along each vertex. You will have three smaller triangular sections. - Take the three angles (the vertices) and arrange them on a flat surface. 5. **Demonstrate the Sum of Angles:** - Position the cuts of the triangle such that the angles are all aligned adjacent to each other. You can place them in a linear fashion or arrange them to form a straight line. #### Demonstration: - Place the first angle (let's say ∠A) next to the second angle (∠B) so that they share a common side. - Then bring the third angle (∠C) to align next to the sum of the other two angles. - The arrangement should demonstrate that when the angles are placed together, they form a straight line. #### Observation: - Measure the angles you labeled in the beginning using the protractor: - For example: - ∠A = 60° - ∠B = 50° - ∠C = 70° - **Calculation:** - Sum of angles = ∠A + ∠B + ∠C = 60° + 50° + 70° = 180°. #### Conclusion: - This visual demonstration shows that the angles at the vertices of the triangle, when rearranged, perfectly fit to form a straight line, confirming that the sum of the angles of a triangle is indeed 180°. #### Application: - Understanding that the sum of angles in a triangle equals 180° is fundamental in various applications such as: - Solving problems in geometry. - Designing buildings and structures. - Creating art that involves triangular shapes. #### Figure: - Here would be a simple illustration unless represented visually (which cannot be done in this text format): ``` C / \ / \ A------B Where: - Angles ∠A, ∠B, ∠C correspondingly represent the angles opposite vertices A, B, and C. ``` This activity not only verifies the geometric principle but also encourages hands-on learning, enabling students to visualize and understand the concept more profoundly. Update (2024-11-19): ### Activity 5: To Verify That the Sum of the Angles of a Triangle is 180° #### **Materials Needed:** 1. Colored paper (various colors) 2. Scissors 3. Ruler 4. Protractor 5. Glue or tape 6. A pencil 7. A straight edge (optional) #### **Method of Construction:** 1. **Draw a Triangle:** - On the colored paper, use the ruler to draw a triangle. Label the triangle as ABC. - Ensure the triangle has different lengths for unique angles (e.g., triangle with sides measuring 5 cm, 6 cm, and 7 cm). 2. **Cut Out the Triangle:** - Carefully cut out the triangle along its edges with scissors. 3. **Label the Angles:** - Use the protractor to measure each angle of the triangle (angle A, angle B, and angle C) and note down the measurements on the triangle with a pencil. - For example, angle A could be 60°, angle B 70°, and angle C 50°. 4. **Cut the Angles:** - Using the scissors, cut out angle A, angle B, and angle C from the triangle separately. Be careful to keep the cuts clean and precise. 5. **Arrange the Angles:** - Take the three angle pieces (A, B, and C) and place them in such a way that they meet at a point (the vertex) forming a straight line. - You can use glue or tape to hold them together at the vertex. #### **Demonstration:** - After arranging the angles together, you should find that they form a straight line. - Use a ruler or a straight edge to demonstrate that the angles line up perfectly to create 180°. #### **Observation (with Measurements):** 1. **Measure Each Angle:** - Measure each of the angles you cut out using the protractor. - Example: - Angle A: 60° - Angle B: 70° - Angle C: 50° 2. **Calculate the Sum:** - Add the three angles together to verify: \[ \text{Sum of angles} = A + B + C = 60° + 70° + 50° = 180° \] 3. **Observation Result:** - Conclude that the sum of the angles of triangle ABC is indeed 180°. #### **Application:** This activity demonstrates the fundamental theorem in geometry that states: "The sum of the angles of a triangle is always equal to 180°." This theorem is crucial in various applications in mathematics and real-world problem-solving such as: 1. **Architecture and Engineering:** - Used when designing triangular structures for stability. 2. **Surveying:** - Helps in calculating land areas and angles in property boundaries. 3. **Triangle-based Calculations in Trigonometry:** - Establishes the basis for determining unknown angles and sides in triangles. ### **Figures:** 1. **Figure 1: Triangle ABC**  2. **Figure 2: Angles Cut Out**  3. **Figure 3: Angles Arranged to Form a Straight Line**  ### Conclusion: Through this paper cutting and pasting method, we have visually and practically verified that the sum of the angles in a triangle is always equal to 180°.