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Tag: MATHS CLASS9TH

  • Let’s proceed with more advanced problems and diagrams to Euclid’s Geometry.

    Advanced Example 1: Proof Using Parallel Lines

    Question

    Prove that the sum of the interior angles on the same side of a transversal is less than 180∘180^\circ if the lines are not parallel.

    Solution

    1. Given: Two lines l1l_1 and l2l_2, and a transversal tt, such that l1l_1 and l2l_2 are not parallel.
    2. To Prove: ∠1+∠2<180∘\angle 1 + \angle 2 < 180^\circ, where ∠1\angle 1 and ∠2\angle 2 are interior angles on the same side of the transversal.
    3. Construction:
      • Extend l1l_1 and l2l_2 indefinitely to meet at point PP.
    4. Proof:
      • Since l1l_1 and l2l_2 are not parallel, they meet at point PP.
      • In △PQT\triangle PQT, the sum of angles of the triangle is 180∘180^\circ: ∠1+∠2+∠P=180∘.\angle 1 + \angle 2 + \angle P = 180^\circ.
      • Here, ∠P\angle P is a positive angle, so: ∠1+∠2<180∘.\angle 1 + \angle 2 < 180^\circ.
    5. Conclusion:
      The sum of interior angles on the same side of a transversal is less than 180∘180^\circ when the lines are not parallel.
    6. Diagram: Let’s plot this graphically.

    Graph Explanation

    • The blue line (l1l_1) and the green line (l2l_2) represent non-parallel lines.
    • The red dashed line is the transversal (tt) intersecting l1l_1 and l2l_2.
    • The purple point is the intersection of l1l_1 and tt, while the orange point is the intersection of l2l_2 and tt.

    From the graph:

    • The interior angles formed on the same side of the transversal tt (between l1l_1 and l2l_2) are less than 180∘180^\circ, confirming the proof.

    Would you like to explore proofs involving parallel lines or other examples?