Question

complete the two - column proof.
given: ( overline{tr}congoverline{ue},overline{te}congoverline{ur} )
prove: ( angle tcongangle u )
third angles theorem
all right angles are congruent.
opposite angles of a rectangle are congruent.
corresponding parts of congruent triangles are congruent.
alternate interior angles are congruent.

1. ( overline{tr}congoverline{ue} )
2. ( overline{te}congoverline{ur} )
3. ( overline{re}congoverline{er} )
4. ( \triangle tercong\triangle ure )
5. ( angle tcongangle u )

Explanation:

Step1: Identify the congruent triangles

Given \(\overline{TR}\cong\overline{UE}\), \(\overline{TE}\cong\overline{UR}\), and \(\overline{RE}\cong\overline{ER}\) (reflexive property). By the SSS (Side - Side - Side) congruence criterion, \(\triangle TER\cong\triangle URE\).

Step2: Use the property of congruent triangles

When two triangles are congruent, their corresponding parts are congruent. So, if \(\triangle TER\cong\triangle URE\), then \(\angle T\) and \(\angle U\) (which are corresponding angles of the congruent triangles \(\triangle TER\) and \(\triangle URE\)) are congruent.

Answer:

Corresponding parts of congruent triangles are congruent.