Class 9 Maths — Parallel Lines (Lines & Angles)

Example 1: In the diagram, \(l \parallel m\). If \(\angle1 = 68^\circ\), find \(\angle5\) and \(\angle3\). Solution: \(\angle5\) corresponds to \(\angle1\) ⇒ \(\angle5 = 68^\circ\). Also, \(\angle3\) is alternate interior to \(\angle5\) ⇒ \(\angle3 = 68^\circ\).
Example 2: A transversal cuts two lines. If a pair of interior angles on the same side is \(120^\circ\) and \(60^\circ\), prove the lines are parallel. Solution: Their sum is \(180^\circ\). By the converse of the co-interior angle property, the lines are parallel.
Example 3: \(l\) and \(m\) are cut by \(t\). If \(\angle2 = 3x+5\) and \(\angle8 = 2x+35\) are alternate exterior angles and lines are parallel, find \(x\) and each angle. Solution: For parallel lines, AEA are equal: \(3x+5 = 2x+35 \Rightarrow x=30\). Then each angle is \(3(30)+5 = 95^\circ\).
Example 4 (Algebraic): If \(\angle3\) and \(\angle6\) are co-interior and \(\angle3 = 5y-10\), \(\angle6 = 3y+20\) with \(l \parallel m\), find \(y\). Solution: Co-interior sum \(=180^\circ\): \((5y-10)+(3y+20)=180 \Rightarrow 8y+10=180 \Rightarrow y=21.25\).

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