Ex.3.1 Q7 Understanding Quadrilaterals Solution - NCERT Maths Class 8

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Question

a) Find \(x +y +z\)

b) Find \(x +y +z +w\)

Text Solution

What is Known?

The sum of the measures of all the interior angles of a quadrilateral is \(360^\circ\) and that of a triangle is \(180^\circ\)

What is Unknown?

Angle\( x, y, z \) and \(w \) in the above figures \(a \) and \(b\)

Reasoning:

The unknown angles can be estimated by using the angle sum property of a quadrilateral and triangle accordingly.

Steps:

(a) Find \(x +y +z\)

Sum of linear pair of angles is\(  = {180^{\rm{o}}}\)

\[\begin{align}x + {90^{\rm{o}}} & = {180^{\rm{o}}}{\text{ (Linear pair)}}\\x & = {180^{\rm{o}}} - {90^{\rm{o}}}\\x & = {90^{\rm{o}}}\end{align}\]

And

\[\begin{align}z + 30^{\rm{o}} &= {180^{\rm{o}}}{\rm{ }}({\text{Linear pair}})\\z & = 180^{\rm{o}} - {30^{\rm{o}}}\\z &= {150^{\rm{o}}}\end{align}\]

And

\[\begin{align}y & = {90^{\rm{o}}} + {30^{\rm{o}}}{\text{ (Exterior angle theorem)}}\\y & = {120^{\rm{o}}}\end{align}\]

\[\begin{align}x + y + z & = {90^{\rm{o}}} + {120^{\rm{o}}} + {150^{\rm{o}}}\\& = {360^{\rm{o}}}\end{align}\]

(b)Find \(x +y +z\)

The sum of the measures of all the interior angles of a quadrilateral is \(360^\circ.\)

Using the angle sum property of a quadrilateral,

Let \(n\) is the fourth interior angle of the quadrilateral.

\[\begin{align}6{{0}^{\text{o}}}+8{{0}^{o}}+12{{0}^{o}}+n &=36{{0}^{o}}  \\26{{0}^{o}}+n& =36{{0}^{o}}  \\n& =36{{0}^{o}}-26{{0}^{o}}  \\n& =10{{0}^{o}}  \\\end{align}\]

sum of linear pair of angles is \(  = {180^{\rm{o}}}\)

\[\begin{align}w + 100^\circ & = 180^\circ {\rm{ }} & \ldots \left( 1 \right)\\x + 120^\circ & = 180^\circ  & \ldots \left( 2 \right)\\y + 80^\circ & = 180^\circ &   \ldots \left( 3 \right)\\z + 60^\circ & = 180^\circ &  \ldots \left( 4 \right)\end{align}\]

Adding equation (\(1), (2), (3) \)and \((4),\)

\[\begin{align}w{\rm{ }} + {\rm{ }}100^\circ + x + 120^\circ + y + 80^\circ + z + 60^\circ & = {\rm{ }}180^\circ +180^\circ + 180^\circ + 180^\circ \\w + x + y + z + 360^\circ &= 720^\circ \\w{\rm{ }} + x + y{\rm{ }} + z &= 720^\circ - 360^\circ \\w + {\rm{ }}x + y + {\rm{ }}z &= 360^\circ\end{align}\]

The sum of the measures of the external angles of any polygon is \(360^\circ \) .

  
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