# Ex.6.2 Q5 Lines and Angles Solution - NCERT Maths Class 9

## Question

In the given figure, if \(AB \| CD,\;\; \angle APQ = 50 ^ { \circ }\) and \(\angle PRD = 127 ^ { \circ }\), find \(x\) and \(y\).

## Text Solution

**What is known?**

\(AB \| CD,\;\; \angle APQ = 50 ^ { \circ }\) and \(\angle PRD = 127 ^ { \circ }\)

**What is unknown?**

\(x =?\) and \(y = ?\)

**Reasoning:**

- When a ray intersects a line, sum of adjacent angles formed is \(180^ {\circ}\).
- When two parallel lines are cut by a transversal, alternate interior angles formed are equal.

**Steps:**

\(AB\) and \(CD\) are parallel lines cut by transversal \(PQ\) hence the alternate interior angles formed are equal.

\(\angle APQ = \angle PQR\) and hence \(x = 50^ {\circ}\).

Similarly, \(AB\) and \(CD\) are parallel lines cut by transversal \(PR\) hence the alternate angles formed are equal.

\[\begin{align} \angle APR+ \angle PRD &= 127 ^ { 0 } \\ \angle APQ + \angle QPR &= \angle PRD \\&= 127 ^ { \circ } \\ 50 ^ { \circ } + y &= 127 ^ { 0 } \\ y &= (127 ^ { 0 } - 50 ^ { \circ }) \\ y &= 77 ^ { \circ } \end{align}\]