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Ex.6.3 Q4 Lines and Angles Solution - NCERT Maths Class 9

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Question

In fig. below, if lines \(PQ\) and \(RS\) intersect at point \(T\), such that \(\angle PRT = 40^\circ \) ,\(\angle RPT = 95^\circ \) and \(\angle TSQ = 75^\circ \) , find \(\angle SQT\) .

Text Solution

What is known?

\(\angle PRT = 40^\circ \), \(\angle RPT = 95^\circ \) and \(\angle TSQ = 75^\circ \)

What is unknown?

\(\angle SQT\)

Reasoning:

As we know when two line intersect each other at a point then there are two pairs of vertically opposite angles formed are equal.

Angle sum property of a triangle:

Sum of the interior angles of a triangle is \(360^\circ\).

Steps:

Given,

\(\angle PRT = 40^\circ \), \(\angle RPT = 95^\circ \) and \(\angle TSQ = 75^\circ \)

In \(\Delta PRT\)

\(\begin{align}\angle PTR + \angle PRT + \angle RPT &= 180^\circ {\rm{ }}\\( \text{Angle sum property of a}&\text { triangle}\rm{.})\\\\\angle PTR + 40^\circ + 95^\circ &= 180^\circ \\\angle PTR &= 180^\circ - 135^\circ\\\angle{PTR} &= 45^\circ {\rm{ }}\end{align}\)

Now,

\(\begin{align}\angle QTS& = \angle PTR\quad\\( \text{Vertically }&\text {opposite angles})\\\\\angle QTS &= 45^\circ\quad \left( {\rm{i}} \right)\end{align}\)

In \(\Delta TSQ\)

\(\begin{align}\angle QTS + \angle TSQ + \angle SQT &= 180^\circ\\( \text{Angle sum property of a }&\text {triangle}.)\\\\45^\circ + 75^\circ + \angle SQT &= 180^\circ \\ [ \text{From }&( \rm{i} ) ]\\\\\angle SQT &= 180^\circ - 120^\circ \\\angle SQT &= 60^\circ \end{align}\)

Hence,\(\angle SQT = 60^\circ {\rm{ }}\)

  
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