Question

In the given figure $AB\parallel CD\parallel EF$, find the angles x and y

Answer 1

Hint:
In the given parallel lines we can see that BC is the transversal of the the parallel lines AB and CD and CF is the transversal of the parallel lines EF and CD and using the properties that alternate interior angles are equal and co interior angles sum upto ${180}^{\circ }$ we can find the value of x and y.

Complete step by step solution:
We are given that $AB\parallel CD\parallel EF$
And from the figure it is clear that BC is a transversal of the parallel lines AB and CD
And CF is a transversal of the parallel lines EF and CD.

Now let's consider the parallel lines AB and CD
We are given that $\mathrm{\angle }ABC={75}^{\circ }$
Using the property , interior opposite angles are equal
Here $\mathrm{\angle }ABC$and $\mathrm{\angle }BCD$are interior opposite angles
Hence they are equal
$$\begin{gathered} \Rightarrow \mathrm{\angle }ABC=\mathrm{\angle }BCD \\ \Rightarrow \mathrm{\angle }ABC=\mathrm{\angle }BCF+\mathrm{\angle }FCD \\ \Rightarrow {75}^{\circ }=y+{25}^{\circ } \\ \Rightarrow {75}^{\circ }-{25}^{\circ }=y \\ \Rightarrow {50}^{\circ }=y \end{gathered}$$
Now let's consider the parallel lines CD and EF
Here we can use the property co interior angles sum upto ${180}^{\circ }$
Here $\mathrm{\angle }EFC$and $\mathrm{\angle }FCD$are co interior angles
$$\begin{gathered} \Rightarrow \mathrm{\angle }EFC+\mathrm{\angle }FCD={180}^{\circ } \\ \Rightarrow x+{25}^{\circ }={180}^{\circ } \\ \Rightarrow x={180}^{\circ }-{25}^{\circ } \\ \Rightarrow x={155}^{\circ } \end{gathered}$$

Hence we get the values of x and y.

Note:
1) When a transversal intersects two parallel lines,
2) The corresponding angles are equal.
3) The vertically opposite angles are equal.
4) The alternate interior angles are equal.
5) The alternate exterior angles are equal.
6) The pair of interior angles on the same side of the transversal is supplementary.