Question

c. given: tr intersects np, tu ≅ ru, and nu ≅ pu
verify that △tun ≅ △rup.

enter the answer in the space provided.

Explanation:

Step1: Identify vertical angles

When two lines intersect, vertical angles are equal. So, \(\angle TUN=\angle RUP\) (vertical angles are congruent).

Step2: List given congruent sides

We are given \(TU\cong RU\) and \(NU\cong PU\).

Step3: Apply the SAS (Side - Angle - Side) congruence criterion

In \(\triangle TUN\) and \(\triangle RUP\), we have \(TU = RU\) (given), \(\angle TUN=\angle RUP\) (vertical angles), and \(NU = PU\) (given). By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

\(\triangle TUN\cong\triangle RUP\) by the SAS (Side - Angle - Side) congruence criterion.