Question

In each of the figures given below, AB || CD. Find the value of x in each case.

Answer 1

(i)

Draw $EF\parallel AB\parallel CD$.
Now, $AB\parallel EF$ and BE is the transversal.
Then,
$$\begin{gathered} \angle ABE=\angle BEF\left[\text{Alternate Interior Angles}\right] \\ \Rightarrow \angle BEF=35° \end{gathered}$$
Again, $EF\parallel CD$ and DE is the transversal.
Then,
$$\begin{gathered} \angle DEF=\angle FED \\ \Rightarrow \angle FED=65° \\ \therefore x°=\angle BEF+\angle FED \\ =\left(35+65\right)° \\ =100° \\ \mathrm{or},x=100 \end{gathered}$$

(ii)

Draw $EO\parallel AB\parallel CD$.
Then, $\angle EOB+\angle EOD=x°$
Now, $EO\parallel AB$ and BO is the transversal.
$$\begin{gathered} \therefore \angle EOB+\angle ABO=180°\left[\text{Consecutive Interior Angles}\right] \\ \Rightarrow \angle EOB+55°=180° \\ \Rightarrow \angle EOB=125° \end{gathered}$$
Again, $EO\parallel CD$ and DO is the transversal.
$$\begin{gathered} \therefore \angle EOD+\angle CDO=180°\left[\text{Consecutive Interior Angles}\right] \\ \Rightarrow \angle EOD+25°=180° \\ \Rightarrow \angle EOD=155° \end{gathered}$$
Therefore,
$$\begin{gathered} x°=\angle EOB+\angle EOD \\ =\left(125+155\right)° \\ =280° \\ \mathrm{or},x=280 \end{gathered}$$

(iii)

Draw $EF\parallel AB\parallel CD$.
Then, $\angle AEF+\angle CEF=x°$
Now, $EF\parallel AB$ and AE is the transversal.
$$\begin{gathered} \therefore \angle AEF+\angle BAE=180°\left[\text{Consecutive Interior Angles}\right] \\ \Rightarrow \angle AEF+116=180 \\ \Rightarrow \angle AEF=64° \end{gathered}$$
Again, $EF\parallel CD$ and CE is the transversal.
$$\begin{gathered} \angle CEF+\angle ECD=180°\left[\text{Consecutive Interior Angles}\right] \\ \Rightarrow \angle CEF+124=180 \\ \Rightarrow \angle CEF=56° \end{gathered}$$

Therefore,
$$\begin{gathered} x°=\angle AEF+\angle CEF \\ =\left(64+56\right)° \\ =120° \\ \mathrm{or},x=120 \end{gathered}$$