Question

which statements can be concluded from the diagram and used to prove that the triangles are similar by the sas similarity theorem? \\(\bigcirc\\) \\(\frac{rs}{vu} = \frac{st}{ut}\\) and \\(\angle s \cong \angle u\\) \\(\bigcirc\\) \\(\frac{rs}{vu} = \frac{st}{ut} = \frac{rt}{vt}\\) \\(\bigcirc\\) \\(\frac{rs}{vu} = \frac{tu}{ts}\\) and \\(\angle s \cong \angle u\\) \\(\bigcirc\\) \\(\frac{rs}{vu} = \frac{tu}{ts} = \frac{rt}{vt}\\)

Explanation:

Step1: Recall SAS Similarity Theorem

The SAS (Side - Angle - Side) similarity theorem states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

Step2: Identify the sides and included angle

• For triangle \(RST\) and triangle \(VUT\):
• The right angles \(\angle S\) and \(\angle U\) are the included angles.
• Calculate the ratios of the sides:
• \(RS = 12\), \(VU=6\), \(ST = 16\), \(UT = 8\)
• \(\frac{RS}{VU}=\frac{12}{6} = 2\)
• \(\frac{ST}{UT}=\frac{16}{8}=2\)
• Also, \(\angle S\cong\angle U\) (both are right angles).

• Now let's check the options:
• Option 1: \(\frac{RS}{VU}=\frac{ST}{UT}\) (since \(\frac{12}{6}=\frac{16}{8} = 2\)) and \(\angle S\cong\angle U\) (both right angles), which satisfies the SAS similarity theorem.
• Option 2: \(\frac{RS}{VU}=\frac{ST}{UT}=\frac{RT}{VT}\) is the SSS (Side - Side - Side) similarity condition, not SAS.
• Option 3: \(\frac{RS}{VU}=\frac{TU}{TS}\) is incorrect because \(\frac{RS}{VU}=\frac{12}{6} = 2\) and \(\frac{TU}{TS}=\frac{8}{16}=\frac{1}{2}\), they are not equal.
• Option 4: \(\frac{RS}{VU}=\frac{TU}{TS}=\frac{RT}{VT}\) is incorrect as the ratios are not equal and it is not the SAS condition.

Answer:

\(\boldsymbol{\frac{RS}{VU}=\frac{ST}{UT}\text{ and }\angle S\cong\angle U}\) (the first option)