91Ó°ÊÓ

In the figure it can be observed that, $\stackrel{¯}{RT}â‰\dots \stackrel{¯}{AS}$and$\stackrel{¯}{RS}â‰\dots \stackrel{¯}{AT}$.

In two right triangles, if corresponding hypotenuse and one pair of corresponding sides are congruent, then by $\mathit{H}\mathit{L}$congruency, both triangles are congruent and in congruent triangles, corresponding angles are always congruent.

First prove the congruency of triangles$\mathrm{Δ}\text{TSR}$and$\mathrm{Δ}\text{TSA}$.

So, by $SSS$congruency $\mathrm{Δ}TSRâ‰\dots \mathrm{Δ}TSA$.

Since, $\mathrm{Δ}TSRâ‰\dots \mathrm{Δ}TSA$, therefore, by congruent part of congruent triangles

i. role="math" localid="1648880107394" $∠\text{SRT}â‰\dots ∠\text{TAS}$…… (a)

II. $∠\text{RST}â‰\dots ∠\text{ATS}$…… (b)

Now, in $\mathrm{Δ}\text{SOR}$and $\mathrm{Δ}\text{TOA}$:

So, by $AAS$congruency $\mathrm{Δ}\text{SOR}â‰\dots \mathrm{Δ}\text{TOA}$.

By congruent part of congruent triangles, $∠\text{RSO}â‰\dots ∠\text{ATO}$ …… (c).

Now, from (b), we have$∠\text{RST}â‰\dots ∠\text{ATS}$.

So,

$∠\text{RSO}+∠\text{OST}â‰\dots ∠\text{ATO}+∠\text{OTS}$

$∠\text{TSO}â‰\dots ∠\text{STO}$[Since, $∠\text{RSO}â‰\dots ∠\text{ATO}$, from (c)]

$∠\text{TSA}â‰\dots ∠\text{STR}$

Thus, by the above proof it is clear that $∠\text{TSA}â‰\dots ∠\text{STR}$.