Question

complete each congruence statement by naming the corresponding angle or side.

1. \\( \triangle vtu \cong \triangle igh \\)
2. \\( \triangle vut \cong \triangle tji \\)

state what additional information is required in order to know that the triangles are congruent
for the reason given.

1. sss
2. asa

\\( \overline { u v } \cong ? \\)
\\( \angle u t v \cong ? \\)

Explanation:

Step1: Analyze triangle congruence for \(\triangle VTU\cong\triangle IGH\)

In congruent triangles, corresponding parts are congruent. For \(\triangle VTU\cong\triangle IGH\), the side \(UV\) in \(\triangle VTU\) corresponds to the side \(IH\) in \(\triangle IGH\). So, \(\overline{UV}\cong\overline{IH}\).

Step2: Analyze triangle congruence for \(\angle UTV\) in \(\triangle VUT\cong\triangle TJI\)

In congruent triangles \(\triangle VUT\cong\triangle TJI\), the angle \(\angle UTV\) corresponds to the angle \(\angle TIJ\). So, \(\angle UTV\cong\angle TIJ\).

Step3: Analyze additional information for SSS (Side - Side - Side) congruence

For SSS congruence, we need all three sides of one triangle to be equal to the corresponding sides of another triangle. In the given figure for problem 7, we already have two pairs of equal sides (\(DJ = JK\) as marked). We need \(LD=LK\) to satisfy the SSS criterion.

Step4: Analyze additional information for ASA (Angle - Side - Angle) congruence

For ASA congruence, we need two angles and the included side of one triangle to be equal to the corresponding two angles and included side of another triangle. In the given figure for problem 8, we already have one pair of equal sides (\(RS = JL\) as marked) and one pair of equal angles. We need \(\angle R=\angle J\) to satisfy the ASA criterion.

Answer:

1. \(\overline{UV}\cong\overline{IH}\)
2. \(\angle UTV\cong\angle TIJ\)
3. \(LD = LK\)
4. \(\angle R=\angle J\)