One-fifth of the area of a triangle $ABC$ is cut off by a line $DE$ drawn parallel to $BC$ such that$D$ is on $AB$ and $E$ is on $AC$. If $BC=10 \mathrm{cm}$, then what is $DE$ equal to?

In the adjoining figure $AB\parallel CD$ and $PQ,$ $QR$ intersect $AB$ and $CD$ at $E, F, G$ and $H$ respectively. Given that $\angle PEB={80}^{\mathrm{o}}$, $\angle QHD={120}^{\mathrm{o}}$ and $\angle PQR=x^{\mathrm{o}}$, find the value of $x.$

In the diagram $AB\parallel GH\parallel DE$ and $GF\parallel BD\parallel HI,$$\angle \mathrm{FGC}={80}^{\mathrm{o}}$. Find the value of $\angle CHI.$

Two parallel lines $AB$ and $CD$ are intersected by a transversal $EF$ at $M$ and $N$ respectively. The lines $MP$ and $NP$ are the bisectors of interior angle $\angle BMN$ and $\angle DNM$ on the same side of the transversal. Then, $\angle MPN$ is equal to

$AB\parallel CD$ and the line $EF$ cuts these lines at the points $M$ and $N$ respectively. Bisectors of angles $\angle BMN$ and $\angle MND$ meet at the point $Q.$ Find the value of $\angle MQN.$

In the given diagram, $\mathrm{TU}\parallel \mathrm{PS}$ and points $\mathrm{Q}$ and $\mathrm{R}$ lie on $\mathrm{PS}$. Also, $\angle \mathrm{PQT}=\mathrm{x}°$, $\angle \mathrm{RQT}=\left(\mathrm{x}-50\right)°$ and $\angle \mathrm{TUR}=\left(\mathrm{x}+25\right)°$. What is the measure of$\angle \mathrm{URS}$?

The value of z (in degrees), in given figure, is

If $\angle 5=(4x-20)°$ and $\angle 7=(x+10)°$, find the value of $\angle 1$.