Question
In figure, $\angle\text{AOC}\cong\angle\text{PYR}$ and $\angle\text{AOC}\cong\angle\text{PYR}$. Name the angle which is congruent to $\angle\text{AOB}.$
✓
Answer
$\angle\text{AOC}\cong\angle\text{PYR}\ ...(\text{i})$
Also, $\angle\text{BOC}\cong\angle\text{QYR}$
Now, $\angle\text{AOC}=\angle\text{AOB}+\angle\text{BOC}$ and $\angle\text{PYR}=\angle\text{PYQ}+\angle\text{QYR}$
By putting the value of $\angle\text{AOC}$ and $\angle\text{PYR}$ in equation (i), we get:
$\angle\text{AOB}+\angle\text{BOC}\cong\angle\text{PYQ}+\angle\text{QYR}$
$\angle\text{AOB}\cong\angle\text{PYQ}$ $(\because \angle\text{BOC}\cong\angle\text{QYR})$
Hence, $\angle\text{AOB}\cong\angle\text{PYQ}$
Also, $\angle\text{BOC}\cong\angle\text{QYR}$
Now, $\angle\text{AOC}=\angle\text{AOB}+\angle\text{BOC}$ and $\angle\text{PYR}=\angle\text{PYQ}+\angle\text{QYR}$
By putting the value of $\angle\text{AOC}$ and $\angle\text{PYR}$ in equation (i), we get:
$\angle\text{AOB}+\angle\text{BOC}\cong\angle\text{PYQ}+\angle\text{QYR}$
$\angle\text{AOB}\cong\angle\text{PYQ}$ $(\because \angle\text{BOC}\cong\angle\text{QYR})$
Hence, $\angle\text{AOB}\cong\angle\text{PYQ}$
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