Question

find the value of x in the figure below

Answer 1

Given: Angles of the triangle = x° , (2x + 2)° and (2x - 7)°.

Need to find: The angle x?

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As we know that,

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$\begin{gathered}\star\:{\underline{\boxed{\frak{Sum \: of \: angles \: of \: triangle_{\:(triangle)} = 180^{\circ}}}}}\\\\\\ \bf{\dag}\:{\underline{\frak{Putting\:given\:values\:in\:formula,}}}\\\\\\ :\implies\sf x^{\circ} + (2x + 2)^{\circ}+(2x - 7)^{\circ} = 180^{\circ} \\\\\\ :\implies\sf 5x - 5^{\circ} = 180^{\circ}\\\\\\ :\implies\sf 5x = 180^{\circ} + 5 ^{\circ} \\\\\\ :\implies\sf 5x = 185 ^{\circ} \\\\\\ :\implies\sf x = \cancel \frac{185}{5}\\\\\\ :\implies{\underline{\boxed{\frak{\purple{x = 37 ^{\circ}}}}}}\:\bigstar\\\\\end{gathered}$

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Therefore,

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$\sf In \: \triangle PQR, \\\\\\ \sf : \implies Angle \: Q, x = 37^{\circ} \\ \\ \\ : \implies \sf Angle \: P, (2x + 2)^{\circ} = (2(37) + 2)^{\circ} = 76^{\circ} \\\\\\ \sf : \implies Angle \: R, (2x - 7)^{\circ} = (2(37) - 7)^{\circ} = 67^{\circ}$

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$\therefore\:{\underline{\sf{Hence,\: The \: value \: of \: x \: in \: the \: triangle \:is\:\bf{37^{\circ}}\sf{respectively}.}}}$

Answer 2

$\Large{\underbrace{\underline{\sf{Understanding\: the\; Question}}}}$

Here in this question, concept of angle sum property is used. We are given a triangle and it's 3 angles. These angles are in term of x and we are asked to find value of x. We can find the value of x, by applying angle sum Property of triangle.

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$\large\underline{\boxed{ \pmb{\maltese\;\frak{ Angle\:Sum\: Property}}}}$

The sum of all angles of triangle measures 180°, no triangle can be formed with sum of angles greater or lesser than 180°.

Now let's solve the problem!!

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In triangle PQR

∠Q+ ∠R+ ∠P=180°  (A.S.P.)

By substituting given values:

$\sf :\implies x+(2x-7^{\circ})+(2x+2^\circ)=180^\circ$

$\sf:\implies x+2x-7^\circ+2x+2^\circ=180^\circ$

$\sf:\implies 5x-5^\circ=180^\circ$

$\sf :\implies 5x=180^\circ+5^\circ$

$\sf :\implies 5x=185^\circ$

$\sf:\implies x=\dfrac{185^\circ}{5}$

$:\implies\large{\underline{\boxed{\sf{\red{ x=\bf{37^{\circ}}}}}}}$

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$\large\underline{\boxed{ \pmb{\maltese\;\frak{Required\:angles\:are:-}}}}$

$\sf :\implies \angle Q=x=\bf 37^{\circ}}$

$\sf :\implies \angle R=\tt{(2x-7)^\circ}$

$\sf :\implies \angle R=\tt{[2(37)-7]^\circ}$

$\sf :\implies \angle R=\tt{[74-7]^\circ}$

$\sf :\implies \angle R=\bf{67^\circ}$

$\sf :\implies \angle P=\tt{[2x+2]^\circ}$

$\sf :\implies \angle P=\tt{[2(37)+2]^\circ}$

$\sf :\implies \angle P=\tt{[74+2]^\circ}$

$\sf :\implies \angle P=\bf{76^\circ}$

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