Question

In fig, PQ||XY and $\angle 1 = {110^0},\angle 8 = {70^0}$, Then find $\angle 3$ and $\angle 6$
A. $\angle 3 = {60^0}$, $\angle 6 = {120^0}$
B. $\angle 3 = {110^0}$, $\angle 6 = {70^0}$
C. $\angle 3 = {100^0}$, $\angle 6 = {80^0}$
D. $\angle 3 = {90^0}$, $\angle 6 = {90^0}$

Answer 1

Hint: According to the question we have to find $\angle 3$ and $\angle 6$ when PQ||XY and $\angle 1 = {110^0},\angle 8 = {70^0}$. According to fig PQ||XY and AB is a transversal line which is cutting a pair of two parallel lines PQ and XY as mentioned above in the figure.
Now we have to use the transversal rule for both of the lines PQ and XY, Where vertically opposite angles are equal to each other.
As in above figure the vertically opposite angles for line PQ are $\angle 1,\angle 3$ and $\angle 2,\angle 4$ similar vertically opposite angles for line XY are $\angle 5,\angle 7$ and $\angle 6,\angle 8$


Complete step by step answer:
Step 1: First of all as mentioned in the question above PQ||XY and $\angle 1 = {110^0}$ So, $\angle 1$ and $\angle 3$ will be vertically opposite angles we can also understand with the help of the diagram mention below,

$ \Rightarrow \angle 1 = \angle 3$……………………………….(1)
Step 2: Now as we know that $\angle 1 = {110^0}$ hence on substituting in the expression (1),
$
   \Rightarrow \angle 1 = \angle 3 \\
   \Rightarrow \angle 3 = {110^0} \\
 $
Step 3: Now, as mentioned in the question above PQ||XY and $\angle 8 = {70^0}$ So, $\angle 6$ and $\angle 8$ will be vertically opposite angles we can also understand with the help of the diagram mention below,

$ \Rightarrow \angle 6 = \angle 8$……………………………….(2)
Step 4: Now as we know that $\angle 8 = {70^0}$ hence on substituting in the expression (2),
$
   \Rightarrow \angle 6 = \angle 8 \\
   \Rightarrow \angle 6 = {70^0} \\
 $
Hence, we have obtained the $\angle 3 = {110^0}$ and $\angle 6 = {70^0}$ with the help of vertically opposite angles rule. Therefore option (B) is correct.

Note: When two lines intersect they form two pairs of opposite angles and vertical angles are always congruent, which means that they are equal to each other.
To angles are said to be supplementary angles when the sum of two angles is ${180^0}$