Question

Find the value of each lettered angle in the figure.
Also state which geometrical fact you used to find the value of the variable.

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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.

Answer 2

$\sf \Large Solution!!$

z + 143° = 180° (Linear pair)

z = 180° - 143°

z = 37°

We can find the value of x by using angle sum property of triangle which is that the sum of the angles in a triangle is 180°.

x + 80° + 52° = 180° (Angle sum property of triangle)

x + 132° = 180°

x = 180° - 132°

x = 48°

Now, to find the value of y, we can either find the value of the trapezium and then find the angle by linear pair as done by @IdyllicAurora or we can directly use the angle sum property of triangle. z, x and y are the interior angles of the smaller triangle. I'll use the angle sum property of triangle to find out y.

x + y + z = 180° (Angle sum property of triangle)

48° + y + 37° = 180°

85° + y = 180°

y = 180° - 85°

y = 95°