If B and Q are acute angles such that B = Q, then prove that B = …

Question

If $ \angle B $ and $\angle Q$ are acute angles such that $\sin B = \sin Q,$ then prove that $ \angle B = \angle Q$.

✓

Answer

Consider two right triangles $ABC$ and $PQR$ in which $ \angle B$ and $\angle Q$ are the right angles.
We have,

In $\triangle ABC$
$\sin B=\frac{AC}{AB}$
and, In $\triangle PQR$
$\sin Q=\frac{PR}{PQ}$
$ \because \quad \sin B = \sin Q$
$ \Rightarrow \quad \frac { A C } { A B } = \frac { P R } { P Q }$
$ \Rightarrow \quad \frac { A C } { P R } = \frac { A B } { P Q } = k($say$) ...... (i)$
$ \Rightarrow AC = kPR$ and $AB = kPQ .....(ii)$
Using Pythagoras theorem in triangles $ABC$ and $PQR,$ we obtain
$A B^2=A C^2+B C^2$ and $\mathrm{PQ}^2=\mathrm{PR}^2+\mathrm{QR}^2$
$ \Rightarrow \quad B C = \sqrt { A B ^ { 2 } - A C ^ { 2 } } \text { and } Q R = \sqrt { P Q ^ { 2 } - P R ^ { 2 } }$
$ \Rightarrow \quad \frac { B C } { Q R } = \frac { \sqrt { A B ^ { 2 } - A C ^ { 2 } } } { \sqrt { P Q ^ { 2 } - P R ^ { 2 } } } = \frac { \sqrt { k ^ { 2 } P Q ^ { 2 } - k ^ { 2 } P R ^ { 2 } } } { \sqrt { P Q ^ { 2 } - P R ^ { 2 } } } [ $using $(ii) ]$
$ \Rightarrow \quad \frac { B C } { Q R } = \frac { k \sqrt { P Q ^ { 2 } - P R ^ { 2 } } } { \sqrt { P Q ^ { 2 } - P R ^ { 2 } } } = k...(iii)$
From $(i)$ and $(iii),$ we get
$ \frac { A C } { P R } = \frac { A B } { P Q } = \frac { B C } { Q R }$
$ \Rightarrow \quad \Delta A C B - \Delta P R Q [$By $S.A.S$ similarity$]$
$ \therefore \quad \angle B = \angle Q$
Hence proved.