Question

1. select the reasons that justify the statements in the proof table:

given: \\( \frac { n o } { q o } = \frac { p o } { m o } \\) prove: \\( \triangle m n o \sim \triangle p q o \\)
\\( \

$$\begin{array} { | c | c | } \\hline \\text { statement } & \\text { reason } \\\\ \\hline \\frac { n o } { q o } = \\frac { p o } { m o } & \\text { a } \\\\ \\hline \\angle n o m \\cong \\angle q o p & \\text { b } \\\\ \\hline \\triangle m n o \\sim \\triangle p q o & \\text { c } \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Identify the reason for the first statement

The first statement \(\frac{NO}{QO}=\frac{PO}{MO}\) is given in the problem, so the reason (A) is "Given".

Step2: Identify the reason for the second statement

\(\angle NOM\cong\angle QOP\) because they are vertical angles. Vertical angles are always congruent. So the reason (B) is "Vertical angles are congruent".

Step3: Identify the reason for the third statement

We have two - sides in proportion (\(\frac{NO}{QO}=\frac{PO}{MO}\)) and the included angle congruent (\(\angle NOM\cong\angle QOP\)). By the Side - Angle - Side (SAS) similarity criterion, \(\triangle MNO\sim\triangle PQO\). So the reason (C) is "SAS (Side - Angle - Side) similarity criterion".

Answer:

A. Given; B. Vertical angles are congruent; C. SAS (Side - Angle - Side) similarity criterion.