question_answer

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, \[x=45{}^\circ \text{ }and\text{ }y=135{}^\circ \]