Question

# To determine:To prove: /_RST ~ /_XYZ by the SAS similarity theorem. Given: The triangles are similar by theorem SAS. /_R=/_X (Given)

Similarity
To determine:To prove: $$\displaystyle\triangle{R}{S}{T}\sim\triangle{X}{Y}{Z}$$ by the SAS similarity theorem.
Given:
The triangles are similar by theorem SAS.
$$\displaystyle\angle{R}=\angle{X}$$ (Given)

2020-12-01
Calculation:
SAS theorem for similarity states that the ratios of two sides of one triangle to the corresponding triangle’s two side must be equal. Angle between equal ratios sides of both the triangle must be equal.
In triangle RST and XYZ,
$$\displaystyle\angle{R}=\angle{X}$$ (Given)
Ratios of the sides with end R and X must be equal.
Consider the ratios of sides as follows:
$$\displaystyle\frac{{{R}{S}}}{{{X}{Y}}}=\frac{{{R}{T}}}{{{X}{Z}}}$$
Hence, theratios of $$\displaystyle\frac{{{R}{S}}}{{{X}{Y}}}{\quad\text{and}\quad}\frac{{{R}{T}}}{{{X}{Z}}}$$ must be equal to prove $$\displaystyle\triangle{R}{S}{T}\sim\triangle{X}{Y}{Z}$$ by SAS similarity theorem.