Question

In figure, AB and CD are parallel lines intersected by a transversal PQ at L and M respectively. If $$\mathrm{\angle }LMD={35}^{\circ }$$ , find $$\mathrm{\angle }ALM$$ and $$\mathrm{\angle }PLA$$.

Answer 1

Hint: Here we will have to apply concepts of –
Sum of all the angles on a straight line is $${180}^{\circ }$$ & Alternate interior angles are equal when two lines are parallel & there is a transversal. Applying this concept of geometry, we can get values of angles asked for in the above question.

Complete step-by-step answer:
Given:
AB and CD are parallel lines (AB II CD) intersected by a transversal PQ at L and M respectively.
[ Transversal – A line that intersects two lines on the same plane intersecting at two different points.]
$$\mathrm{\angle }LMD={35}^{\circ }$$
To find : $$\mathrm{\angle }ALM$$ & $$\mathrm{\angle }PLA$$
In this figure, $$\mathrm{\angle }ALM$$ and $\mathrm{\angle }LMD$ are anterior interior angles.
[ Anterior internal angles are a pair of angles on the inner side of each side of each of those two lines but on the opposite sides of the transversal]
We know that anterior interior angles are equal when two lines are parallel & there is a transversal.
AB and CD are parallel lines, AB II CD & PQ is transversal.
$$\therefore \mathrm{\angle }ALM=\mathrm{\angle }LMD={35}^{\circ }$$
Hence, the value of $$\mathrm{\angle }ALM$$ is $${35}^{\circ }$$.
Now to find value of $$\mathrm{\angle }PLA$$,
We need to know that, Sum of all the angles on a straight line is $${180}^{\circ }$$ .
In the above figure, on line segment LM,
$$\mathrm{\angle }ALM$$+ $$\mathrm{\angle }PLA$$ = $${180}^{\circ }$$
$\Rightarrow \mathrm{\angle }PLA={180}^{\circ }-\mathrm{\angle }ALM$
$={180}^{\circ }-{35}^{\circ }$
$$={145}^{\circ }$$
$\therefore \mathrm{\angle }PLA={145}^{\circ }$
Hence the value of $$\mathrm{\angle }PLA$$ is ${145}^{\circ }$

Note: Alternate interior angles are equal, when lines are parallel to each other & there is a transversal so by implementing this concept we can get a linear equation to be solved carefully to get the ultimate answer.