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# Find the values of the unknowns $x$ and $y$ in the following diagrams:"

Given: Triangles in the above-given diagram with unknown angles $x$ and $y$.

To do: To find the values of unknown angles $x$ and $y$ in each case.

Solution:

For convenience, we name all the triangles given in the diagram as $\triangle ABC$.

$(i).\ \angle y+\angle ACD=180^{\circ}$       [linear pair]

$\Rightarrow \angle y\ +\ 120^{\circ}=180^{\circ}$

$\Rightarrow \angle y=60^{\circ}$

By using the angle sum property

In triangle $ABC$

$\angle A+\angle B+\ \angle C=180^{\circ}$      [$\angle A=x,\ \angle C=y$]

$\Rightarrow x+50^{\circ}+60^{\circ}=180^{\circ}$

$\Rightarrow x=180^{\circ}-110^{\circ}$

$\Rightarrow x=70^{\circ}$

$(ii).\ \angle EAD=\angle y$          [vertically opposite angles are the same]

$80^{\circ}=\angle y$

By using the angle sum property

In triangle $ABC$

$\angle A+\angle B+\angle C=180^{\circ}$

$\Rightarrow 80^{\circ}+\ 50^{\circ}\ +x=180^{\circ}$

$\Rightarrow x=180^{\circ}-130^{\circ}$

$\Rightarrow x=50^{\circ}$

$(iii)$ By using the angle sum property

In triangle $ABC$

$\angle A+\angle B+\angle C=180^{\circ}$

$\Rightarrow 50^{\circ}+60^{\circ}+y=180^{\circ}$

$\Rightarrow y=180^{\circ}-110^{\circ}$

$\Rightarrow y=70^{\circ}$

We know that, $x+y=180^{\circ}$             [linear pair]

$\Rightarrow x+\ 70^{\circ}=180^{\circ}$

$\Rightarrow x=110^{\circ}$

$(iv).\ x=\angle DCE$          [vertically opposite angles are the same]

$60^{\circ}=x$

By using the angle sum property

In trianfle $ABC$,

$\angle A+\angle B+\angle C=180^{\circ}$

$\Rightarrow 30^{\circ}+60^{\circ}+y=180^{\circ}$

$\Rightarrow y=180^{\circ}-90^{\circ}$

$\Rightarrow y=90^{\circ}$

$(v).\ \angle EAD=y=90^{\circ}$                 [vertically opposite angles are the same]

In triangle $ABC$,

By using the angle sum property,

$\angle A+\angle B+\angle C=180^{\circ}$

$\Rightarrow x+x+90^{\circ}=180^{\circ}$

$\Rightarrow 2x=180^{\circ}-90^{\circ}$

$\Rightarrow 2x=90^{\circ}$

$\Rightarrow x=\frac{90^{\circ}}{2}=45^{\circ}$

$(vi)$. by using the angle sum property,

$y=x=\angle A=\angle B=\angle C$     [vertically opposite angles are the same]

In triangle $ABC$

$\Rightarrow x+x+x=180^{\circ}$

$\Rightarrow 3x=180^{\circ}$

$\Rightarrow x=\frac{180^{\circ}}{3}$

$\Rightarrow x=60^{\circ}$

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Updated on: 10-Oct-2022

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