In the diagram below, PQRST is a trapezium made up of a parallelogram PQRS and an isosceles triangle PST where PT = PS. It is given that RS = a, RQ = b and SR = ST.

1. Identify equal sides and the included equal angle needed for congruence

Since $PQRS$ is a parallelogram,
$QR = PS \quad \text{and} \quad QR \parallel PS.$
Given that
$RS = ST.$
Since $R, S, T$ are collinear and $QR \parallel PS$,
$\angle QRS = \angle PST.$

2. Apply triangle congruence and conclude the required equality

In $\triangle QRS$ and $\triangle PST$,
$QR = PS, \quad RS = ST, \quad \angle QRS = \angle PST.$
Therefore,
$\triangle QRS \cong \triangle PST \quad \text{by SAS}.$
Hence, corresponding sides are equal:
$QS = PT.$

1. Identify the first isosceles triangle using the given equal sides

Given that $PT=PS$, $riangle PST$ is isosceles.

2. Use part (a) to prove the middle triangle is isosceles

From part (a), $QS=PT$.

Since $PT=PS$, it follows that
$QS=PS.$
Hence $riangle PQS$ is isosceles.

3. Use opposite sides of the parallelogram to prove the third triangle is isosceles

Since $PQRS$ is a parallelogram, opposite sides are equal, so
$QR=PS.$
But $PS=PT$ and $PT=QS$, so
$QR=QS.$
Hence $riangle QRS$ is isosceles.