The given figure shows a parallelogram PQRS. ST and QU are the bisectors of ∠PSR and ∠PQR respectively.

If SU = 4 cm and QU = 10 cm, then what is the value of (ST + TQ)?

In a parallelogram, opposite angles are equal.

Therefore, in parallelogram PQRS,

∠PSR = ∠PQR

$$\begin{gathered} \Rightarrow \frac{1}{2}\angle \mathrm{PSR}=\frac{1}{2}\angle \mathrm{PQR} \\ \Rightarrow \angle \mathrm{TSU}=\angle \mathrm{TQU} \end{gathered}$$

In a parallelogram, opposite sides are parallel.

∴PQ||SR

⇒ TQ||SU

∴ ∠TQU + ∠QUS = 180° [Sum of the interior angles on the same side of transversal is 180°]

∴∠TSU + ∠QUS = 180°

⇒ TS||QU  [If sum of the interior angles on the same side of transversal is 180°, then the lines are parallel]

Thus, in quadrilateral TQUS, TQ||SU and TS||QU.

∴TQUS is a parallelogram.

ST = QU = 10 cm

TQ = SU = 4 cm

∴ST + TQ = 10 cm + 4 cm = 14 cm

Thus, the value of (ST + TQ) is 14 cm.