Question
In the figure given alongside, AB || CD, EF || BC, $\angle\text{BAC}=60^\circ$ and $\angle\text{DHE}=50^\circ.$ Find $\angle\text{GCH}$ and $\angle\text{AGH}.$
✓
Answer
AB || CD and AC is the transversal.$\Rightarrow\angle\text{BAC}=\angle\text{ACD}=60^\circ$ (alternate angles)
i. e. $\angle\text{BAC}=\angle\text{GCH}=60^\circ$ Now, $\angle\text{DHF}=\angle\text{CHG}=50^\circ$ (vertically opposite angles) In $\triangle\text{GCH},$ by angle sum property,$\angle\text{GCH}+\angle\text{CHG}+\angle\text{CGH}=180^\circ$
$\Rightarrow60^\circ+50^\circ\angle\text{CGH}=180^\circ$
$\Rightarrow\angle\text{CGH}=70^\circ$
Now, $\angle\text{CGH}+\angle\text{AGH}=180^\circ$ (linear pair)$\Rightarrow70^\circ+\angle\text{AGH}=180^\circ$
i. e. $\angle\text{BAC}=\angle\text{GCH}=60^\circ$ Now, $\angle\text{DHF}=\angle\text{CHG}=50^\circ$ (vertically opposite angles) In $\triangle\text{GCH},$ by angle sum property,$\angle\text{GCH}+\angle\text{CHG}+\angle\text{CGH}=180^\circ$
$\Rightarrow60^\circ+50^\circ\angle\text{CGH}=180^\circ$
$\Rightarrow\angle\text{CGH}=70^\circ$
Now, $\angle\text{CGH}+\angle\text{AGH}=180^\circ$ (linear pair)$\Rightarrow70^\circ+\angle\text{AGH}=180^\circ$
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