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.

$\angle POS$ is the angle subtended by arc PS at the center O. $\angle PQS$ is the angle subtended by arc PS at the circumference.

Given $\angle PQS = x$:

Reason: Angle at center is twice the angle at the circumference subtended by the same arc.

2.1.2 $\angle R$

The angle at R refers to $\angle PRS$.

$\angle PRS$ and $\angle PQS$ are both angles subtended by the same arc PS at the circumference.

Given $\angle PQS = x$:

Reason: Angles in the same segment are equal.

2.1.3 $\angle T_1$

From the diagram, $T_1$ refers to $\angle PTS$.

We are given $\angle PQS = x$, so $\angle TQP = x$.
Since $PQ \parallel SR$, and PR is a transversal, $\angle QPR = \angle PRS$ (alternate interior angles).
From 2.1.2, $\angle PRS = x$, so $\angle QPR = x$. Thus, $\angle TPQ = x$.

$\angle PTS$ is an exterior angle to $\triangle PTQ$.

Reason: Exterior angle of a triangle is equal to the sum of the two interior opposite angles.

2.2 Prove that $\triangle TSR$ is an isosceles triangle.

From 2.1.2, we know $\angle TRS = \angle PRS = x$.
Since $PQ \parallel SR$, and QS is a transversal, $\angle QSR = \angle PQS$ (alternate interior angles).
Given $\angle PQS = x$, so $\angle QSR = x$. Thus, $\angle TSR = x$.

We have $\angle TRS = x$ and $\angle TSR = x$. Since $\angle TRS = \angle TSR$, the sides opposite these angles must be equal.

Therefore, $TS = TR$, which means $\triangle TSR$ 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.

From 2.1.1, we found $\angle POS = 2x$.
From 2.1.3, we found $\angle PTS = 2x$.

We observe that $\angle POS = \angle PTS = 2x$. Both angles are subtended by the chord PS.

From the diagram, points O and T lie on the same side of the chord PS.

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).