1. Problem 4: Prove that \(\triangle ONP \cong \triangle PQO\) given \(\angle N\) and \(\angle Q\) are right angles and \(NO = PQ\). 2. Since \(\angle N\) and \(\angle Q\) are right angles, triangles \(ONP\) and \(PQO\) are right triangles by definition. 3. The triangles share side \(NO = PQ\) by given. 4. By the Reflexive Property, side \(NP\) is common to both triangles. 5. Using the RHS (Right angle-Hypotenuse-Side) congruence criterion for right triangles, \(\triangle ONP \cong \triangle PQO\). 1. Problem 5: Prove that \(\triangle SRT \cong \triangle UTR\) given \(ST \parallel RU\) and \(SR \parallel TU\). 2. Since \(ST \parallel RU\), alternate interior angles \(\angle SRT \cong \angle UTR\). 3. Since \(SR \parallel TU\), alternate interior angles \(\angle TSR \cong \angle TUT\) (or corresponding angles). 4. Side \(RT\) is common to both triangles. 5. By the ASA (Angle-Side-Angle) congruence criterion, \(\triangle SRT \cong \triangle UTR\). 1. Problem 6: Prove that \(\triangle VWX \cong \triangle ZYX\) given \(\angle W\) and \(\angle Y\) are right angles, \(VX \cong ZX\), and \(X\) is the midpoint of \(WY\). 2. Since \(\angle W\) and \(\angle Y\) are right angles, triangles \(VWX\) and \(ZYX\) are right triangles. 3. Given \(VX \cong ZX\). 4. Since \(X\) is midpoint of \(WY\), segments \(WX \cong XY\). 5. By the RHS congruence criterion, \(\triangle VWX \cong \triangle ZYX\).