Question

1. select the postulate or theorem that you can use to conclude that the triangles are similar. aa similarity postulate sas similarity theorem sss similarity theorem

Explanation:

Step1: Identify side - length ratios

In \(\triangle GMK\) and \(\triangle PMS\), we have \(\frac{GM}{PM}=\frac{3}{4}\) and \(\frac{KM}{SM}=\frac{2}{6}=\frac{1}{3}\). Since \(\frac{GM}{PM}
eq\frac{KM}{SM}\), we are not using SSS similarity. Also, we have no information about angles to use AA similarity.
We note that \(\angle GMK\) and \(\angle PMS\) are vertical angles, so \(\angle GMK=\angle PMS\). And we have the ratio of the sides including these angles: \(\frac{GM}{PM}=\frac{3}{4}\) and \(\frac{KM}{SM}=\frac{2}{6}=\frac{1}{3}\).

Step2: Apply 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 two triangles are similar. Here, the included angles \(\angle GMK\) and \(\angle PMS\) are congruent (vertical angles) and we have side - side ratios. So we use the SAS similarity theorem.

Answer:

SAS Similarity Theorem