Question

t is the midpoint of \\( \overline{qs} \\) and \\( \overline{pr} \\). complete the proof that \\( \triangle pst \cong \triangle rqt \\).

Explanation:

Step1: Midpoint Definition

By the definition of a midpoint, if \(T\) is the midpoint of a segment, it divides the segment into two congruent parts. So, for statement 3 (\(\overline{QT}\cong\overline{ST}\)), the reason is "Definition of midpoint" (since \(T\) is the midpoint of \(\overline{QS}\)). For statement 4 (\(\overline{PT}\cong\overline{RT}\)), the reason is also "Definition of midpoint" (since \(T\) is the midpoint of \(\overline{PR}\)).

Step2: Vertical Angles Theorem

For statement 5 (\(\angle PTS\cong\angle QTR\)), the reason is "Vertical angles are congruent". Vertical angles are formed when two lines intersect. In this case, the lines \(PS\) and \(QR\) (or the lines containing \(\overline{PR}\) and \(\overline{QS}\)) intersect at \(T\), and \(\angle PTS\) and \(\angle QTR\) are vertical angles.

Step3: SAS Congruence Criterion

For statement 6 (\(\triangle PST\cong\triangle RQT\)), the reason is "Side - Angle - Side (SAS) Congruence Theorem". In \(\triangle PST\) and \(\triangle RQT\), we have \(\overline{PT}\cong\overline{RT}\) (from step 1), \(\angle PTS\cong\angle QTR\) (from step 2), and \(\overline{ST}\cong\overline{QT}\) (from step 1). The SAS theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

1. Given
2. Given
3. Definition of midpoint
4. Definition of midpoint
5. Vertical angles are congruent
6. Side - Angle - Side (SAS) Congruence Theorem