Determine the following in terms of x with reasons. 2.1.1 POS 2.1.2 R 2.1.3 T1 Prove that triangle TSR is an isosceles triangle. Prove that the points P, O, T and S form a cyclic quadrilateral.

Here are the solutions to the geometry problem: 2.1 Determine the following in terms of x with reasons. 2.1.1 POS Step 1: Identify the relationship between POS and PQS. POS is the angle subtended by arc PS at the center O. PQS is the angle subtended by arc PS at the circumference. Step 2: Apply the angle at center theorem. POS = 2 PQS Given PQS = x: POS = 2x Reason: Angle at center is twice the angle at the circumference subtended by the same arc. POS = 2x 2.1.2 R Step 1: Identify the angle at R. The angle at R refers to PRS. Step 2: Apply the angles in the same segment theorem. PRS and PQS are both angles subtended by the same arc PS at the circumference. PRS = PQS Given PQS = x: PRS = x Reason: Angles in the same segment are equal. R = x 2.1.3 T_1 Step 1: Identify the angle T_1. From the diagram, T_1 refers to PTS. Step 2: Find the angles in PTQ. We are given PQS = x, so TQP = x. Since PQ SR, and PR is a transversal, QPR = PRS (alternate interior angles). From 2.1.2, PRS = x, so QPR = x. Thus, TPQ = x. Step 3: Apply the exterior angle theorem for PTQ. PTS is an exterior angle to PTQ. PTS = TQP + TPQ PTS = x + x PTS = 2x Reason: Exterior angle of a triangle is equal to the sum of the two interior opposite angles. T_1 = 2x 2.2 Prove that TSR is an isosceles triangle. Step 1: Find the angles in TSR. From 2.1.2, we know TRS = PRS = x. Since PQ SR, and QS is a transversal, QSR = PQS (alternate interior angles). Given PQS = x, so QSR = x. Thus, TSR = x. Step 2: Compare the angles in TSR. We have TRS = x and TSR = x. Since TRS = TSR, the sides opposite these angles must be equal. Step 3: Conclude that TSR is isosceles. Therefore, TS = TR, which means TSR is an isosceles triangle. Reason: Sides opposite equal angles are equal. 2.3 Prove that the points P, O, T and S form a cyclic quadrilateral. Step 1: Identify relevant angles involving points P, O, T, S. From 2.1.1, we found POS = 2x. From 2.1.3, we found PTS = 2x. Step 2: Compare these angles and their relationship to chord PS. We observe that POS = PTS = 2x. Both angles are subtended by the chord PS. Step 3: Check the position of O and T relative to chord PS. From the diagram, points O and T lie on the same side of the chord PS. Step 4: Apply the converse of the angles in the same segment theorem. If two points (O and T) on the same side of a line segment (PS) subtend equal angles at the endpoints of the segment, then the four points (P, O, T, S) are concyclic. Therefore, P, O, T, S form a cyclic quadrilateral. Reason: Angles subtended by the same chord (PS) at points on the same side of the chord are equal (converse of angles in the same segment). Send me the next one 📸