1.

Find the values of x y z in the following figure​​

Answer» <html><body><p><strong>Step-by-step explanation:</strong></p><p></p><p>As per the figure,</p><p></p><p>⠀⠀⠀★ ∠1 = x</p><p>⠀⠀⠀★ ∠2 = x/2</p><p>⠀⠀⠀★ ∠3 = z</p><p>⠀⠀⠀★ ∠4 = y</p><p></p><p>We are asked to calculate the value of <strong>x</strong><strong>,</strong><strong>y</strong><strong>,</strong><strong>z</strong>. Clearly, <strong>∠1</strong><strong> </strong>and <strong>∠</strong><strong>3</strong><strong> </strong>; <strong>∠</strong><strong>2</strong> and <strong>∠</strong><strong>4</strong> are vertically opposite angles. Since, the vertically opposite angles equal, so ∠1 and ∠3 ; ∠2 and ∠4 will be equal.</p><p></p><p>Thus, we can say that :</p><p></p><p>➝ ∠1 = ∠3 ⇒ x</p><p>➝ ∠2 = ∠4 ⇒ x/2</p><p></p><ul><li>Value of <strong>y</strong> can be written as <strong>x</strong><strong>/</strong><strong>2</strong><strong>.</strong></li><li>Value of <strong>z</strong><strong> </strong>can be written as <strong>x</strong><strong>.</strong></li></ul><p></p><p><strong>The</strong><strong> </strong><strong>sum</strong><strong> </strong><strong>of</strong><strong> </strong><strong>all</strong><strong> </strong><strong>these</strong><strong> </strong><strong>angles</strong><strong> </strong><strong>will</strong><strong> </strong><strong>be</strong><strong> </strong><strong>3</strong><strong>6</strong><strong>0</strong><strong>°</strong><strong> </strong><strong>as</strong><strong> </strong><strong>they</strong><strong> </strong><strong>are</strong><strong> </strong><strong>formin</strong><strong>g</strong><strong> </strong><strong>a</strong><strong> </strong><strong>complete</strong><strong> </strong><strong>angle</strong><strong>.</strong> <a href="https://interviewquestions.tuteehub.com/tag/writing-25453" style="font-weight:bold;" target="_blank" title="Click to know more about WRITING">WRITING</a> it in the <a href="https://interviewquestions.tuteehub.com/tag/form-996208" style="font-weight:bold;" target="_blank" title="Click to know more about FORM">FORM</a> of a linear <a href="https://interviewquestions.tuteehub.com/tag/equation-974081" style="font-weight:bold;" target="_blank" title="Click to know more about EQUATION">EQUATION</a>,</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360^\circ }} \\" class="latex-formula" id="TexFormula1" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7B%5Cangle%201%20%2B%20%5Cangle%202%20%2B%20%5Cangle%203%20%2B%20%5Cangle%204%20%3D%20360%5E%5Ccirc%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360^\circ }} \\"/></p><p></p><p>Substitute the measure angles.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {x+ \dfrac{x}{2} + z + y = 360^\circ }} \\" class="latex-formula" id="TexFormula2" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7Bx%2B%20%5Cdfrac%7Bx%7D%7B2%7D%20%2B%20z%20%2B%20y%20%3D%20360%5E%5Ccirc%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {x+ \dfrac{x}{2} + z + y = 360^\circ }} \\"/></p><p></p><p>Now, substitute the expression of <strong>y</strong> and <strong>z</strong> which have been found using the property of vertically opposite angles.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {x+ \dfrac{x}{2} + x + \dfrac{x}{2} = 360^\circ }} \\" class="latex-formula" id="TexFormula3" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7Bx%2B%20%5Cdfrac%7Bx%7D%7B2%7D%20%2B%20x%20%2B%20%5Cdfrac%7Bx%7D%7B2%7D%20%3D%20360%5E%5Ccirc%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {x+ \dfrac{x}{2} + x + \dfrac{x}{2} = 360^\circ }} \\"/></p><p></p><p>Taking the LCM and solving further.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {\dfrac{2x + x + 2x + x}{2} = 360^\circ }} \\" class="latex-formula" id="TexFormula4" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7B%5Cdfrac%7B2x%20%2B%20x%20%2B%202x%20%2B%20x%7D%7B2%7D%20%3D%20360%5E%5Ccirc%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {\dfrac{2x + x + 2x + x}{2} = 360^\circ }} \\"/></p><p></p><p>Performing addition in the numerator of the <a href="https://interviewquestions.tuteehub.com/tag/fraction-458259" style="font-weight:bold;" target="_blank" title="Click to know more about FRACTION">FRACTION</a> in the LHS.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {\dfrac{6x}{2} = 360^\circ }} \\" class="latex-formula" id="TexFormula5" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7B%5Cdfrac%7B6x%7D%7B2%7D%20%3D%20360%5E%5Ccirc%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {\dfrac{6x}{2} = 360^\circ }} \\"/></p><p></p><p>Transposing 2 from L.H.S. to R.H.S.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {6x= 360^\circ \times 2 }} \\" class="latex-formula" id="TexFormula6" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7B6x%3D%20360%5E%5Ccirc%20%5Ctimes%202%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {6x= 360^\circ \times 2 }} \\"/></p><p></p><p>Performing multiplication in RHS.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {6x= 720^\circ }} \\" class="latex-formula" id="TexFormula7" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7B6x%3D%20720%5E%5Ccirc%20%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {6x= 720^\circ }} \\"/></p><p></p><p>Transposing 6 from L.H.S. to R.H.S.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \sf{\quad {x= \cancel{ \dfrac{720^\circ}{6}} }} \\" class="latex-formula" id="TexFormula8" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Csf%7B%5Cquad%20%7Bx%3D%20%5Ccancel%7B%20%5Cdfrac%7B720%5E%5Ccirc%7D%7B6%7D%7D%20%20%7D%7D%20%5C%5C%20" title="\longrightarrow \sf{\quad {x= \cancel{ \dfrac{720^\circ}{6}} }} \\"/></p><p></p><p>Dividing 720 by 6.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \quad\underline{\boxed { \textbf{\textsf{x = 120}}^\circ }}\\" class="latex-formula" id="TexFormula9" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Cquad%5Cunderline%7B%5Cboxed%20%7B%20%5Ctextbf%7B%5Ctextsf%7Bx%20%3D%20120%7D%7D%5E%5Ccirc%20%7D%7D%5C%5C%20" title="\longrightarrow \quad\underline{\boxed { \textbf{\textsf{x = 120}}^\circ }}\\"/></p><p></p><p>Now, we have to <a href="https://interviewquestions.tuteehub.com/tag/find-11616" style="font-weight:bold;" target="_blank" title="Click to know more about FIND">FIND</a> the value of other <strong>three </strong><strong>angles</strong>.</p><p></p><p><strong>Value</strong><strong> </strong><strong>of</strong><strong> </strong><strong>∠2</strong> : x/2 = 120°/2 = <strong>6</strong><strong>0</strong><strong>°</strong></p><p></p><p><img align="absmiddle" alt="\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{2 = 60}}^\circ }}\\" class="latex-formula" id="TexFormula10" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Cquad%5Cunderline%7B%5Cboxed%20%7B%20%5Cangle%5Ctextbf%7B%5Ctextsf%7B2%20%3D%2060%7D%7D%5E%5Ccirc%20%7D%7D%5C%5C%20" title="\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{2 = 60}}^\circ }}\\"/></p><p></p><p><strong>V</strong><strong>a</strong><strong>l</strong><strong>u</strong><strong>e</strong><strong> </strong><strong>of </strong><strong>∠</strong><strong>3</strong> : Same as the value of <strong>x</strong>, since the vertically opposite angles are equal.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{3 = 120}}^\circ }}\\" class="latex-formula" id="TexFormula11" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Cquad%5Cunderline%7B%5Cboxed%20%7B%20%5Cangle%5Ctextbf%7B%5Ctextsf%7B3%20%3D%20120%7D%7D%5E%5Ccirc%20%7D%7D%5C%5C%20" title="\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{3 = 120}}^\circ }}\\"/></p><p></p><p><strong>Value</strong><strong> </strong><strong>o</strong><strong>f</strong><strong> </strong><strong>∠</strong><strong>4</strong><strong> : </strong>Same as the value of <strong>x</strong><strong>/</strong><strong>2</strong>, since the vertically opposite angles are equal.</p><p></p><p><img align="absmiddle" alt="\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{4 = 60}}^\circ }}\\" class="latex-formula" id="TexFormula12" src="https://tex.z-dn.net/?f=%20%20%5Clongrightarrow%20%5Cquad%5Cunderline%7B%5Cboxed%20%7B%20%5Cangle%5Ctextbf%7B%5Ctextsf%7B4%20%3D%2060%7D%7D%5E%5Ccirc%20%7D%7D%5C%5C%20" title="\longrightarrow \quad\underline{\boxed { \angle\textbf{\textsf{4 = 60}}^\circ }}\\"/></p><p></p><p><img align="absmiddle" alt="\underline{ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad} \\" class="latex-formula" id="TexFormula13" src="https://tex.z-dn.net/?f=%20%5Cunderline%7B%20%5Cqquad%5Cqquad%5Cqquad%5Cqquad%5Cqquad%5Cqquad%5Cqquad%5Cqquad%7D%20%5C%5C%20" title="\underline{ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad} \\"/></p><p></p><p><strong><u>Therefore</u></strong><strong><u>,</u></strong></p><p></p><p>⠀⠀⠀★ <strong>∠1</strong> = 120°</p><p>⠀⠀⠀★ <strong>∠2 </strong>= 60°</p><p>⠀⠀⠀★<strong> ∠3 </strong>= 120°</p><p>⠀⠀⠀★ <strong>∠4</strong> = 60°</p></body></html>


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