# Ex.6.2 Q3 Lines and Angles Solution - NCERT Maths Class 9

## Question

In the given figure, if \(AB \,‖\, CD\), \(EF \,\bot \,CD\) and \(\angle GED = 126^ {\circ},\) find \(\angle AGE, \; \angle GEF\) and \(\angle FGE.\)

## Text Solution

**What is known?**

\(AB ‖ CD\), \(\text{EF} \bot \text{CD}\) and \(\angle GED = 126^ {\circ},\)

**What is unknown?**

\(\angle AGE =?,\;\; \angle GEF=?\) and \(\angle FGE =?\)

**Reasoning:**

- When two lines intersect, adjacent angles formed are supplementary.
- When two parallel lines are cut by a transversal, alternate interior angles formed are equal.

**Steps:**

Let \( \angle AGE = x,\;\; \angle GED= y\) and \( \angle FGE = z\).

From the figure, we can see that,

\[\begin{align} \angle GED& = \angle GEF + \angle FED \\ y & = (126 ^ { \circ } - 90 ^ { \circ }) \\ \angle GEF & = y = 36 ^ { \circ } \end{align}\]

\(AB\) and \(CD\) are parallel lines cut by a transversal, the pair of alternative angles formed are equal.

\[\begin{align} \angle AGE &= \angle GED \\ \angle AGE &= x = 126 ^ { \circ } \end{align}\]

Line \(AB\) is intersected by line \(GE\) hence adjacent angles formed are supplementary.

\[\begin{align} x + z &= 180 ^ { \circ } \\ 126 ^ { \circ } + z &= 180 ^ { \circ } \\ z &= 180 ^ { \circ } - 126 ^ {\circ } \\ \quad &= 54 ^ {\circ} \\ \angle FGE = z &= 54 ^ { \circ } \end{align}\]