To use the Angle-Side-Angle (ASA) congruence theorem to prove that \(\triangle CES\) is congruent to \(\triangle RTS\), Steve needs to prove that two angles and the included side of one triangle are equal to the corresponding parts of the other triangle.
The choices provided are:
- \(CS = RS\)
- \(ES = TS\)
- \(TR = EC\)
- \(E = T\)
To satisfy the ASA condition, he needs to show that the angles surrounding the included side are congruent, along with one pair of corresponding sides being equal.
Assuming that he already has two angles known or proven to be congruent between the two triangles, he will need to find the length of the side that is included between those two angles equal in both triangles.
If we consider the options provided, the best option would likely be one that represents the validation of the included sides between two equal angles.
Looking at these options, if he knows the included side between the two angles (for example if angles C and E are one pair and angles S and T another), he would ideally need the information about the side that is shared between these angles.
Since we don't have details on which angles are known, we cannot definitively say which among the provided options to choose based on the angle information that might already be available.
If CS and RS represent the sides opposite the angles, then option 1: \(CS = RS\) can imply that if \(CE\) (an included side, for example) is known, he is closer to using ASA.
Therefore, Steve needs "CS = RS" at minimum for ASA congruence, provided he already has two angle congruences established.
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