# To prove : The similarity of \triangle NRT with respect to \triangle NS

To prove : The similarity of $$\displaystyle\triangle{N}{R}{T}$$ with respect to $$\displaystyle\triangle{N}{S}{P}$$.
Given information: Here, we have given that $$\displaystyle\overline{{{S}{P}}}$$ is altitude to $$\displaystyle\overline{{{N}{R}}}\ {\quad\text{and}\quad}\ \overline{{{R}{T}}}$$ is altitude to $$\displaystyle\overline{{{N}{S}}}$$.

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Margot Mill
Proof: As, $$\displaystyle\overline{{{S}{P}}}$$ is altitude to $$\displaystyle\overline{{{N}{R}}}$$,
$$\displaystyle\Rightarrow\angle{S}{P}{N}={90}^{{\circ}}$$
Similarity, as $$\displaystyle\overline{{{R}{T}}}$$ is altitude to $$\displaystyle\overline{{{N}{S}}}$$,
$$\displaystyle\Rightarrow\angle{R}{T}{N}={90}^{{\circ}}$$
$$\displaystyle\Rightarrow\angle{R}{T}{N}\stackrel{\sim}{=}\angle{S}{P}{N}$$
Now, In $$\displaystyle\triangle{N}{R}{T}\ {\quad\text{and}\quad}\ \triangle{N}{S}{P}$$
$$\displaystyle\angle{R}{T}{N}\stackrel{\sim}{=}\angle{S}{P}{N}$$ (Proved above)
$$\displaystyle\angle{N}\stackrel{\sim}{=}\angle{N}$$ (Common)
$$\displaystyle\Rightarrow\triangle{N}{R}{T}\sim\triangle{N}{S}{P}$$ (By AA Similarity Rule)