# Ex.6.1 Q2 Lines and Angles Solution - NCERT Maths Class 9

## Question

In the given figure, lines \(XY\) and \(MN\) intersect at \(O.\) If \(\angle POY = 90^ {\circ}\) and \(a:b = 2:3,\) find \(c.\)

## Text Solution

**What is known?**

\(\angle POY= 90^ {\circ}\) and \(a:b = 2:3\)

**What is unknown?**

\(\angle XON = c =?\)

**Reasoning:**

If two lines intersect each other, then the vertically opposite angles formed are equal.

**Steps:**

Line \(OP\) is perpendicular to line \(XY.\) Hence \(\angle POY = \angle POX = 90^ {\circ}\)

\(\begin{align}\angle POX & = \angle POM + \angle MOX\end{align}\)

\[\begin{align} 90 ^ { \circ } & = a + b \quad \ldots \ldots . ( 1 ) \end{align}\]

Since \(a\) and \(b\) are in the ratio \(2:3\) that is, \(a = 2x\) and \(b = 3x \quad \ldots \ldots . (2)\)

Substituting (\(2\)) in (\(1\)),

\[\begin{align} a + b &= 90 ^ { \circ } \\ 2 x + 3 x &= 90 ^ { \circ } \\ 5 x &= 90 ^ { \circ } \\ x = \frac { 90 } { 5 } &= 18 ^ { \circ } \\ a = 2 x &= 2 \times 18 ^ { \circ } \\ a &= 36 ^ { \circ } \\ b = 3 x &= 3 \times 18 ^ { \circ } \\ b &= 54 ^ { \circ } \end{align}\]

Also,\(\begin{align}\angle MOY & = \angle MOP+ \angle POY\end{align}\)

\[\begin{align} & = a + 90 ^ { \circ } \\ & = 36 ^ { \circ } + 90 ^ { \circ } \\&= 126 ^ { \circ } \end{align}\]

Lines \(MN\) and \(XY\) intersect at point \(O\) and the vertically opposite angles formed are equal.

\[\begin{align}\angle XON &= \angle MOY \\ c &=126^ {\circ} \end{align} \]