Question

uv || st. complete the proof that △suv ≅ △ust.

statement
1 uv || st
2 uv ≅ st
3 ∠suv ≅ ∠tsu
4 su ≅ su
5 △suv ≅ △ust

reason
given
given

Explanation:

Step1: Alternate interior angles

Since \( \overline{UV}\parallel\overline{ST}\), by the alternate - interior angles theorem, when two parallel lines are cut by a transversal (\(\overline{SU}\) in this case), \( \angle SUV\cong\angle TSU\).

Step2: Reflexive property

For any segment \( \overline{SU}\), \( \overline{SU}\cong\overline{SU}\) by the reflexive property of congruence (a segment is congruent to itself).

Step3: SAS congruence criterion

We have \( \overline{UV}\cong\overline{ST}\) (given), \( \angle SUV\cong\angle TSU\) (alternate - interior angles from parallel lines), and \( \overline{SU}\cong\overline{SU}\) (reflexive property). By the Side - Angle - Side (SAS) congruence criterion, \( \triangle SUV\cong\triangle UST\).

Answer:

The reason for \( \angle SUV\cong\angle TSU\) is "Alternate - interior angles theorem (because \( \overline{UV}\parallel\overline{ST}\))"; the reason for \( \overline{SU}\cong\overline{SU}\) is "Reflexive property of congruence"; the reason for \( \triangle SUV\cong\triangle UST\) is "SAS (Side - Angle - Side) congruence criterion".