Question

given: \\( \overline { t r } \parallel \overline { c k } \\); u is the midpoint of \\( \overline { r c } \\) prove: \\( \triangle t r u \cong \triangle k c u \\)

Explanation:

Step1: Given Information

The problem states \(\overline{TR}\parallel\overline{CK}\) and \(U\) is the mid - point of \(\overline{RC}\). This is the starting point for the proof.

Step2: Mid - point Property

Since \(U\) is the mid - point of \(\overline{RC}\), by the definition of a mid - point, \(RU = CU\).

Step3: Alternate Interior Angles

Because \(\overline{TR}\parallel\overline{CK}\), the alternate interior angles \(\angle RTU\cong\angle CKU\). (If two parallel lines are cut by a transversal, alternate interior angles are congruent)

Step4: Vertical Angles

\(\angle RUT\cong\angle CUK\) as they are vertical angles.

Step5: ASA Congruence

In \(\triangle TRU\) and \(\triangle KCU\), we have \(\angle RTU\cong\angle CKU\), \(RU = CU\), and \(\angle RUT\cong\angle CUK\). By the Angle - Side - Angle (ASA) congruence criterion, \(\triangle TRU\cong\triangle KCU\).

Answer:

1. Given
2. \(RU = CU\)
3. Alternate Interior Angles Theorem
4. \(\angle RUT\cong\angle CUK\)
5. ASA (Angle - Side - Angle)