Question

3 pipes take 60 min to water the field. how much it will take to water the field with 6 pipes ?direct proportion or inverse proportion
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Answer 1

★ Concept :-

Here the concept of Total Angle of Quadrilateral and Linear Pair of angles has been used. Firstly we can find the fourth angle of trapezium. Then we can apply Linear Pair of Angles to find the required angles using Angle Sum Property of Triangle.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\pink{Sum\;of\;all\;Angles\;of\;Quadrilateral\;=\;\bf{360^{\circ}}}}}$

$\\\;\boxed{\sf{\pink{Sum\;of\;all\;angles\;on\;line\;=\;\bf{180^{\circ}}}}}$

$\\\;\boxed{\sf{\pink{Sum\;of\;all\;angles\;of\;Triangle\;=\;\bf{180^{\circ}}}}}$

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★ Solution :-

Given,

» First angle of Trapezium = 80°

» Second angle of Trapezium = 52°

» Third angles of Trapezium = 143°

Let fourth angle of Trapezium be A°

We know that,

$\\\;\sf{\rightarrow\;\;Sum\;of\;all\;Angles\;of\;Quadrilateral\;=\;\bf{360^{\circ}}}$

By applying values, we get

$\\\;\sf{\rightarrow\;\;80^{\circ}\;+\;52^{\circ}\;+\;143^{\circ}\;+\;A^{\circ}\;=\;\bf{360^{\circ}}}$

$\\\;\sf{\rightarrow\;\;275^{\circ}\;+\;A^{\circ}\;=\;\bf{360^{\circ}}}$

$\\\;\sf{\rightarrow\;\;A^{\circ}\;=\;\bf{360^{\circ}\;-\;275^{\circ}}}$

$\\\;\bf{\rightarrow\;\;A^{\circ}\;=\;\bf{\green{85^{\circ}}}}$

Hence, Fourth Angle of Trapezium = 85°

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~ For value of x ::

We know that Fourth angle of Trapezium and angle y are Linear Pair of Angles. So, applying the formula, we get

$\\\;\sf{\Longrightarrow\;\;Sum\;of\;all\;angles\;on\;line\;=\;\bf{180^{\circ}}}$

By applying values, we get

$\\\;\sf{\Longrightarrow\;\;85^{\circ}\;+\;y\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\Longrightarrow\;\;y\;=\;\bf{180^{\circ}\;-\;85^{\circ}}}$

$\\\;\bf{\Longrightarrow\;\;y\;=\;\bf{\blue{95^{\circ}}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\:\angle\:y\;=\;\bf{\purple{95^{\circ}}}}}}$

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~ For the value of z ::

We see that the Third Angle of Trapezium and angle z are in linear pair with each other. So applying the formula, we get

$\\\;\sf{\Longrightarrow\;\;Sum\;of\;all\;angles\;on\;line\;=\;\bf{180^{\circ}}}$

By applying values, we get

$\\\;\sf{\Longrightarrow\;\;143^{\circ}\;+\;\angle\:z\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\Longrightarrow\;\;\angle\:z\;=\;\bf{180^{\circ}\;-\;143^{\circ}}}$

$\\\;\bf{\Longrightarrow\;\;\angle\:z\;=\;\bf{\orange{37^{\circ}}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\:\angle\:z\;=\;\bf{\purple{37^{\circ}}}}}}$

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~ For the value of x ::

We know that sum of all angles of Triangle is equal to 180° . Then applying the formula, we get

$\\\;\sf{\mapsto\;\;Sum\;of\;all\;angles\;of\;Triangle\;=\;\bf{180^{\circ}}}$

By applying values, we get

$\\\;\sf{\mapsto\;\;\angle\:x\;+\;\angle\:y\;+\;\angle\:z\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\mapsto\;\;\angle\:x\;+\;95^{\circ}\;+\;37^{\circ}\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\mapsto\;\;\angle\:x\;+\;132^{\circ}\;=\;\bf{180^{\circ}}}$

$\\\;\sf{\mapsto\;\;\angle\:x\;=\;\bf{180^{\circ}\;-\;132^{\circ}}}$

$\\\;\bf{\mapsto\;\;\angle\:x\;=\;\bf{\red{48^{\circ}}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\:\angle\:x\;=\;\bf{\purple{48^{\circ}}}}}}$

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★ More to know :-

• Alternative Interior Angles : These are equal angles which are made on opposite sides of transversal between two parallel lines.

• Vertically Opposite Angles : These are the angles formed between two bisectors which are equal and opposite to each other.