Question

1. given: \\(\overline{qr} \cong \overline{ts}, \overline{qr} \parallel \overline{ts}\\) prove: \\(\triangle qrt \cong \triangle tsq\\) image of a quadrilateral q r t s with q to r and s to t marked as equal, and q to s and s to t with an arrow and a mark, and s to t labeled 5, and a diagonal from q to t

Explanation:

Step1: Identify Given Information

We know \(\overline{QR} \cong \overline{TS}\) and \(\overline{QR} \parallel \overline{TS}\), also \(\overline{QT}\) is a common side (or we can use alternate interior angles).

Step2: Find Alternate Interior Angles

Since \(\overline{QR} \parallel \overline{TS}\) and \(\overline{QT}\) is a transversal, \(\angle QRT \cong \angle STQ\) (Alternate Interior Angles Theorem).

Step3: List Congruent Parts

• \(\overline{QR} \cong \overline{TS}\) (Given)
• \(\angle QRT \cong \angle STQ\) (Alternate Interior Angles)
• \(\overline{QT} \cong \overline{QT}\) (Reflexive Property of Congruence)

Step4: Apply SAS Congruence

By Side - Angle - Side (SAS) Congruence Postulate, \(\triangle QRT \cong \triangle TSQ\)

Now filling the table:

Statements	Reasons
2) \(\angle QRT \cong \angle STQ\)	2) Alternate Interior Angles Theorem (since \(\overline{QR} \parallel \overline{TS}\) and \(\overline{QT}\) is transversal)
3) \(\overline{QT} \cong \overline{QT}\)	3) Reflexive Property of Congruence
4) \(\triangle QRT \cong \triangle TSQ\)	4) SAS Congruence Postulate (using \(\overline{QR} \cong \overline{TS}\), \(\angle QRT \cong \angle STQ\), \(\overline{QT} \cong \overline{QT}\))

Answer:

The completed table is as follows:

Statements	Reasons
2) \(\angle QRT \cong \angle STQ\)	2) Alternate Interior Angles Theorem
3) \(\overline{QT} \cong \overline{QT}\)	3) Reflexive Property
4) \(\triangle QRT \cong \triangle TSQ\)	4) SAS Congruence Postulate

And we have proven \(\triangle QRT \cong \triangle TSQ\) using SAS.