# Ex.6.2 Q1 Lines and Angles Solution - NCERT Maths Class 9

## Question

In the given figure, find the values of \(x\) and \(y\) and then show that \(AB ‖ CD\).

## Text Solution

**Reasoning:**

- When two lines intersect, vertically opposite angles formed are equal.
- Also, when a ray intersects a line sum of adjacent angles formed is \(180^ {\circ}\).
- If a transversal intersects two lines such that a pair of alternate angles is equal, then the two lines are parallel to each other.

**Steps:**

Line \(CD\) is intersected with line \(P\), hence the vertically opposite angles so formed are equal. \(y = 130^ {\circ}.\)

Similarly, line \(AB\) is intersected by line \(P\) hence the sum of adjacent angles formed is \(180^ {\circ}.\)

\[\begin{align} x + 50 ^ { \circ } & = 180 ^ { \circ } \\ x & = 180 ^ { \circ } - 50 ^ { \circ } \\ & = 130 ^ { \circ } \end{align}\]

We know that, if a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel. Here we can see that the pair of alternate angles formed when lines \(AB\) and \(CD\) are intersected by transversal \(P\) are equal. Therefore, \(x = y = 130^ {\circ}.\) So we can say the two lines \(AB\) and \(CD\) are parallel. Hence \(AB ‖ CD\) is proved.