Question

complete the two - column proof.
given: \\( \overline { t r } \cong \overline { u e }, \overline { t e } \cong \overline { u r } \\)
prove: \\( \angle t \cong \angle 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 { t r } \cong \overline { u e } \\)
2. \\( \overline { t e } \cong \overline { u r } \\)
3. \\( \overline { r e } \cong \overline { e r } \\)
4. \\( \delta t e r \cong \delta u r e \\)
5. \\( \angle t \cong \angle u \\)

Explanation:

Step1: Given information

We are given that \(\overline{TR}\cong\overline{UE}\) (Statement 1) and \(\overline{TE}\cong\overline{UR}\) (Statement 2).

Step2: Reflexive property

For Statement 3, \(\overline{RE}\cong\overline{ER}\) by the reflexive property of congruence (a segment is congruent to itself).

Step3: SSS congruence

For Statement 4, in \(\triangle TER\) and \(\triangle URE\), we have \(\overline{TR}\cong\overline{UE}\), \(\overline{TE}\cong\overline{UR}\), and \(\overline{RE}\cong\overline{ER}\). By the Side - Side - Side (SSS) congruence criterion, \(\triangle TER\cong\triangle URE\).

Step4: CPCTC

For Statement 5, since \(\triangle TER\cong\triangle URE\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\angle T\cong\angle U\).

Answer:

Corresponding parts of congruent triangles are congruent.