Find angles x and y in each figure. |
(a) |
(b) |
Answer:
(a) Since, in a triangle an exterior angle and the interior adjacent angle form a linear pair, therefore, \[\angle ACB+120{}^\circ =180{}^\circ \] \[\angle ACB=180{}^\circ 120{}^\circ =60{}^\circ \] Since, Δ ABC is isosceles with AB = AC \[\therefore \angle B=\angle C\] \[\angle B=60{}^\circ \,\,\,i.e.,\text{ }y=60{}^\circ \] By the angle sum property, we have \[\angle A+\angle B+\angle C=180{}^\circ \Rightarrow x+y+60{}^\circ \] \[=180{}^\circ \] \[x+60{}^\circ +60{}^\circ =180{}^\circ \Rightarrow x+120{}^\circ =180{}^\circ \] \[x=180{}^\circ 120{}^\circ =60{}^\circ \] Hence, \[x=60{}^\circ \text{ }and\text{ }y=60{}^\circ \] (b) Since, Δ ABC is isosceles with AB = AC \[\therefore \angle B\text{ }=\angle C\Rightarrow \angle B\text{ }=\text{ }x\] Also, \[\angle B+\angle C=90{}^\circ \] \[x+x=90{}^\circ \text{ }[\because \angle B=x]\] \[2x=90{}^\circ \Rightarrow x=45{}^\circ \] But, \[\angle B+y=180{}^\circ \] \[x+y=180{}^\circ \text{ }[\because \angle B=x]\] \[y=180{}^\circ 45{}^\circ \text{ }\left[ \because x=45{}^\circ \right]\] y = 135° Hence, \[x=45{}^\circ \text{ }and\text{ }y=135{}^\circ \]
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