Question

In the figure, find the values of $x$ and $y$ and then show that $AB\parallel CD$.

Answer 1

Hint: In this question, we need to determine the values of $x$ and $y$ and then show that $AB\parallel CD$. Here, we will find the values of $x$ and $y$ using the concept that the sum of linear pairs of angles are equal to ${180}^{\circ }$ and vertically opposite angles are equal. Then, by using the theorem if a transversal intersects two lines such that pairs of alternate interior angles are equal, then lines are parallel we will show $AB\parallel CD$.

Complete step by step answer:
In the given figure, a transversal intersects two lines $AB$ and $CD$.
We know that the sum of linear pairs of angles is equal to ${180}^{\circ }$. Here, the angle ${50}^{\circ }$ and $\mathrm{\angle }X$ are in the linear pair.
Therefore, we have,
${50}^{\circ }+x={180}^{\circ }$
$x={180}^{\circ }-{50}^{\circ }$
$x={130}^{\circ }$
We know that vertically opposite angles are equal. Here, ${130}^{\circ }$ and $\mathrm{\angle }Y$ are vertically opposite angles.
Therefore, we have,
$y={130}^{\circ }$
Hence, $\mathrm{\angle }x=\mathrm{\angle }y$
Therefore, alternate interior angles are equal.
From the theorem we know that if a transversal intersects two lines such that the pair of alternate interior angles are equal, then the lines are parallel.
$\therefore AB\parallel CD$
Hence, $\mathrm{\angle }x=\mathrm{\angle }y={130}^{\circ }$ and $AB\parallel CD$.

Note: In the question it is important to note that, the linear pair is a pair of angles that share a side and a base. Alternately, they are the two angles created along the line when two lines intersect. Here, the angle ${50}^{\circ }$ and $\mathrm{\angle }X$ are in the linear pair. And the sum of linear pairs of angles is equal to ${180}^{\circ }$. Then, vertically opposite angles are the angles opposite to each other when the two lines cross. Here, ${130}^{\circ }$ and $\mathrm{\angle }Y$ are vertically opposite angles.