# Ex.6.1 Q4 Lines and Angles Solution - NCERT Maths Class 9

## Question

In the given figure, if \(x+y = w+z,\) then prove that \(AOB\) is a line.

## Text Solution

**What is known?**

\(\begin{align}x+y = w+z\end{align}\)

**What is unknown?**

To prove that \(AOB\) is a line.

**Reasoning:**

If the sum of two adjacent angles is \(180^ {\circ}\), then the non–common arms of the angles form a line.

**Steps:**

From the figure we can see that:

\(\begin{align}(x + y) +( w + z) = 360^ {\circ} \\ (\text {complete angle})\end{align}\)

It is given that \((x + y) = (w + z),\) Hence

\((x + y) + (w + z) = 360^ {\circ}\) can be written as \((x + y) + (x +y) = 360^ {\circ}\)

\[\begin{align} 2 x + 2 y &= 360 ^ { \circ } \\ 2 ( x + y ) &= 360 ^ { \circ } \\ x + y &= \frac { 360 } { 2 } \\&= 180 ^ { \circ } \end{align}\]

Since sum of adjacent angles \(x\) and \(y\) with \(OA\) and \(OB\) as the non- common arms is \(180^ {\circ}\) we can say that \(AOB\) is a line.