# Ex.6.1 Q3 Lines and Angles Solution - NCERT Maths Class 9

## Question

In the given figure, \(\angle PQR = \angle PRQ\) then prove that \(\angle PQS = \angle PRT.\)

## Text Solution

**What is known?**

\(\angle PQR = \angle PRQ\)

**What is unknown?**

To prove \(\angle PQS = \angle PRT\)

**Reasoning:**

If a ray stands on a line, then the sum of adjacent angles formed is \(180^ {\circ}.\)

**steps:**

Let \(\angle PQR= \angle PRQ = a\)

Let \(\angle PQS = b \) and \(\angle PRT = c \)

Line \(ST \) and \(PQ\) intersect at point \(Q\), then the sum of adjacent angles \(\angle PQS\) and \(\angle PQR\) is \(180^ {\circ}.\)

\(\begin{align} \angle PQS + \angle PQR & = 180 ^ { \circ } \\ b + a & = 180 ^ { \circ } \\ b & = 180 ^ { \circ } - a\ldots . ( 1 ) \end{align}\)

Line \(ST\) and \(PR\) intersect at point \(R\), then the sum of adjacent angles \(\angle PRQ\) and \(\angle PRT \) is \(180^ {\circ}.\)

\(\begin{align} \angle PRQ + \angle PRT & = 180 ^ { \circ } \\ a + c & = 180 ^ { \circ } \\ c & = 180 ^ { \circ } - a\ldots . ( 2 ) \end{align}\)

From equations (\(1\)) and (\(2\)), it is clear that \(b = c.\)

Hence \(\angle PQS = \angle PRT\) is proved.