Question

1. suppose ( overline{tm} cong overline{gl} ) and ( angle m cong angle g ). what additional information is needed to prove ( \triangle mtd cong \triangle gls ) by sas? ( overline{md} cong overline{sg} ) ( overline{td} cong overline{sl} ) ( angle t cong angle l ) ( angle t cong angle s )

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

The SAS congruence criterion 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.
In \(\triangle MTD\) and \(\triangle GLS\), we already know that \(\overline{TM}\cong\overline{GL}\) (given as one side) and \(\angle M\cong\angle G\) (given as the included angle).

Step2: Identify the required side

For the SAS criterion, we need the other side adjacent to the given angle.
In \(\triangle MTD\), the side adjacent to \(\angle M\) is \(\overline{MD}\). In \(\triangle GLS\), the side adjacent to \(\angle G\) is \(\overline{SG}\). But this is not the correct pair.
The correct pair of sides: In \(\triangle MTD\), the sides are \(\overline{TM}\), \(\angle M\), \(\overline{TD}\); in \(\triangle GLS\), the sides are \(\overline{GL}\), \(\angle G\), \(\overline{SL}\)
Since \(\overline{TM}\cong\overline{GL}\) and \(\angle M\cong\angle G\), we need \(\overline{TD}\cong\overline{SL}\)

Answer:

\(\overline{TD}\cong\overline{SL}\) (the second option)