In the given figure, AB || CD and EF || GH. Find the values of x, y, z and t.
Question:

In the given figure, AB || CD and EF || GH. Find the values of xyz and t.

Solution:

In the given figure,

$x=60^{\circ}$ [Vertically-Opposite Angles]

$\angle P R Q=\angle S Q R$ [Alternate Angles]

$y=60^{\circ}$

$\angle A P R=\angle P Q S \quad$ [Corresponding Angles]

$\Rightarrow 110^{\circ}=\angle P Q R+60^{\circ} \quad[\because \angle P Q S=\angle P Q R+\angle R Q S]$

$\Rightarrow \angle P Q R=50^{\circ}$

$\angle P Q R+\angle R Q S+\angle B Q S=180^{\circ} \quad[$ Since AB is a straight line $]$

$\Rightarrow 50^{\circ}+60^{\circ}+z=180^{\circ}$

$\Rightarrow 110^{\circ}+z=180^{\circ}$

$\Rightarrow z=70^{\circ}$

$\angle D S H=z \quad[$ Corresponding Angles $]$

$\Rightarrow \angle D S H=70^{\circ}$

$\therefore \angle D S H=t \quad[$ Vertically-Opposite Angles $]$

$\Rightarrow t=70^{\circ}$

$\therefore x=60^{\circ}, y=60^{\circ}, z=70^{\circ}$ and $t=70^{\circ} .$