Question

complete the proof.
given: \\( \angle spt \cong \angle opt, \overline{sp} \cong \overline{op} \\)
prove: \\( \angle s \cong \angle o \\)

Explanation:

Step1: Recall triangle congruence postulates

• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.
• SAS (Side - Angle - Side): Two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
• SSS (Side - Side - Side): Three sides of one triangle are congruent to three sides of another triangle.
• AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle.

Step2: Analyze given information

We are given that \(\angle SPT\cong\angle OPT\) (an angle), \(\overline{SP}\cong\overline{OP}\) (a side), and \(\overline{PT}\cong\overline{PT}\) (a side, by reflexive property). The angle \(\angle SPT\) is between \(\overline{SP}\) and \(\overline{PT}\), and the angle \(\angle OPT\) is between \(\overline{OP}\) and \(\overline{PT}\). So we have two sides and the included angle congruent.

So, by the SAS (Side - Angle - Side) Postulate, \(\triangle SPT\cong\triangle OPT\).

Answer:

SAS Postulate (Steps 1, 2)