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Question # To prove:The extended proportions that are needed to prove /_RST ~ /_XYZ by the SSS similarity theorem. Given /_RST, /_XYZ are two triangles.

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ANSWERED To prove:The extended proportions that are needed to prove $$\displaystyle\triangle{R}{S}{T}\sim\triangle{X}{Y}{Z}$$ by the SSS similarity theorem.
Given $$\displaystyle\triangle{R}{S}{T},\triangle{X}{Y}{Z}$$ are two triangles. 2021-02-27
Calculation:
SSS theorem for similarity states that the ratios of sides of one triangle to the corresponding triangle’s sides must be equal.
In triangle RST and XYZ,
Consider the ratios of sides as follows:
$$\displaystyle\frac{{{R}{S}}}{{{X}{Y}}}=\frac{{{S}{T}}}{{{Y}{Z}}}=\frac{{{R}{T}}}{{{X}{Z}}}$$
Hence, theratios of $$\displaystyle\frac{{{R}{S}}}{{{X}{Y}}},\frac{{{S}{T}}}{{{Y}{Z}}}{\quad\text{and}\quad}\frac{{{R}{T}}}{{{X}{Z}}}$$ must be equal to prove $$\displaystyle\triangle{R}{S}{T}\sim\triangle{X}{Y}{Z}$$ by SSS similarity theorem.