Question

1. what additional information is needed to prove that the two triangles are congruent by sss?

a (overline{hg}congoverline{vt})
b (overline{hf}congoverline{vt})
c (angle gcongangle v)
d (angle fcongangle t)

1. what additional information is needed to prove that the two triangles are congruent by sss?

a (overline{sn}congoverline{mn})
b (overline{ln}congoverline{ln})
c (angle mcongangle s)
d (angle lnscongangle lnm)

Explanation:

1.

Step1: Recall SSS (Side - Side - Side) congruence criterion

SSS states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
We already know \(GV = TU\) and \(GF=TV\) (from the markings on the sides). For SSS, we need the third pair of sides to be congruent.

Step2: Analyze the options

• Option A: \(\overline{HG}\cong\overline{VT}\) is not relevant as \(VT\) is a side of \(\triangle TVU\) and \(HG\) is a side of \(\triangle GFH\).
• Option B: \(\overline{HF}\cong\overline{VU}\) would give us three pairs of congruent sides (\(GV = TU\), \(GF = TV\), \(HF=VU\)) for the SSS criterion.
• Option C: \(\angle G\cong\angle V\) is an angle - side relationship and is used for SAS (Side - Angle - Side) not SSS.
• Option D: \(\angle F\cong\angle T\) is also an angle - side relationship and is not relevant for SSS.

Step1: Recall SSS (Side - Side - Side) congruence criterion

SSS requires three pairs of congruent sides. We know \(LN = LN\) (common side) and \(LS = LM\) (from the markings on the sides).

Step2: Analyze the options

• Option A: \(\overline{SN}\cong\overline{MN}\) would give us three pairs of congruent sides (\(LN = LN\), \(LS = LM\), \(SN = MN\)) for the SSS criterion.
• Option B: \(\overline{LN}\cong\overline{LN}\) is just stating the common side (already considered in the basic structure, not an additional needed side for SSS when we are looking for a non - common side).
• Option C: \(\angle M\cong\angle S\) is an angle - side relationship and is used for SAS (Side - Angle - Side) not SSS.
• Option D: \(\angle LNS\cong\angle LNM\) is also an angle - side relationship and is not relevant for SSS.

Answer:

B. \(\overline{HF}\cong\overline{VU}\)

2.