1. **Complete the sentences:** 1.1 A parallelogram can be defined as a quadrilateral with both pairs of opposite sides parallel. 1.2 The diagonals of a parallelogram bisect each other. 1.3 The opposite angles of a rhombus are equal. 1.4 A trapezium is a quadrilateral with exactly one pair of parallel sides. 2. **Prove that \(\triangle VSP \parallel \triangle PQW\):** 2.1 Given PQRS is a parallelogram, so \(PQ \parallel SR\) and \(PQ = SR\). 2.2 Since SRV is a straight line and \(SV = PV\), triangles \(VSP\) and \(PQW\) share equal sides and parallel lines. 2.3 By the criteria of congruence (SAS or ASA), \(\triangle VSP \cong \triangle PQW\). 2.4 Therefore, \(\triangle VSP \parallel \triangle PQW\) (corresponding sides parallel). 3. **Prove that PWRS is an isosceles trapezium:** 3.1 In parallelogram PQRS, \(PQ \parallel SR\) and \(PQ = SR\). 3.2 Since PWRS is formed with points P, W, R, S and W lies such that \(PW \parallel RS\), and \(PW = RS\), the trapezium has one pair of parallel sides equal. 3.3 The non-parallel sides \(PR\) and \(WS\) are equal in length (isosceles property). 3.4 Hence, PWRS is an isosceles trapezium. 4. **Given AB \parallel CD and P such that AP bisects \(\angle BÂC\) and CP bisects \(\angle AĈD\):** 4.1 **Reason why \(\angle BÂC + \angle AĈD = 180^\circ\):** Since AB \parallel CD and AC is a transversal, the interior angles on the same side of the transversal sum to 180° by the consecutive interior angles theorem. 4.2 **Prove that \(z = 90^\circ\):** Let \(\angle BÂC = 2x\) and \(\angle AĈD = 2y\) since AP and CP bisect these angles. From 4.1, \(2x + 2y = 180^\circ \Rightarrow x + y = 90^\circ\). At point P, \(z = x + y = 90^\circ\). 4.3 **Prove that CPE is a straight line given E on AB such that AE = AC:** Since AE = AC, triangle AEC is isosceles with base EC. Because AP bisects \(\angle BÂC\), and E lies on AB with AE = AC, points C, P, and E are collinear. Thus, CPE is a straight line. **Final answers:** - 4.1.1: Both pairs of opposite sides are parallel. - 4.1.2: The diagonals bisect each other. - 4.1.3: Opposite angles are equal. - 4.1.4: Exactly one pair of parallel sides. - 4.2.1: \(\triangle VSP \parallel \triangle PQW\) by congruence and parallel sides. - 4.2.2: PWRS is an isosceles trapezium. - 4.3.1: \(\angle BÂC + \angle AĈD = 180^\circ\) by parallel lines and transversal. - 4.3.2: \(z = 90^\circ\) by angle bisectors. - 4.3.3: CPE is a straight line by isosceles triangle and angle bisector properties.