Question 18

When two parallel lines are cut by a transversal, interior angles on the same side add up to 180 degrees.

Step 1 — Identify the lines and angles

The diagram shows two lines, $l$ and $m$. Arrows on lines $l$ and $m$ mean they are parallel. Line $t$ cuts across lines $l$ and $m$. Line $t$ is called a transversal. We are given that $\angle 3$ is 50 degrees. We need to find a relation between $\angle 3$ and $\angle 6$.

Step 2 — Find the relation between $\angle 3$ and $\angle 6$

Angles $\angle 3$ and $\angle 6$ are interior angles. They are on the same side of the transversal $t$. Since lines $l$ and $m$ are parallel, these angles are supplementary. Supplementary angles add up to 180 degrees. So, we can write the relation: $\angle 3 + \angle 6 = 180^\circ$

$\boxed{\text{The relation is } \angle 3 + \angle 6 = 180^\circ.}$ Let us use the given value for $\angle 3$ to check this. We know that $\angle 3 = \mathbf{50^\circ}$. Let us substitute this value into the relation. $50^\circ + \angle 6 = 180^\circ$ Now, we solve for $\angle 6$. $\angle 6 = 180^\circ - 50^\circ$ $\angle 6 = \mathbf{130^\circ}$

Step 3 — Justify the relation

We will prove that $\angle 3 + \angle 6 = 180^\circ$ always holds. This is true when lines $l$ and $m$ are parallel. First, let us look at $\angle 1$ and $\angle 3$. They are vertically opposite angles. Vertically opposite angles are always equal. So, we can write: $\angle 1 = \angle 3$ Next, let us look at $\angle 1$ and $\angle 5$. They are corresponding angles. Since lines $l$ and $m$ are parallel, corresponding angles are equal. So, we can write: $\angle 1 = \angle 5$ From the two statements above, we know this. $\angle 3$ is equal to $\angle 1$. And $\angle 1$ is equal to $\angle 5$. This means $\angle 3$ is also equal to $\angle 5$. So, we have: $\angle 3 = \angle 5$ Now, let us look at $\angle 5$ and $\angle 6$. These two angles form a linear pair on line $m$. Angles in a linear pair add up to 180 degrees. So, we can write: $\angle 5 + \angle 6 = 180^\circ$ We found earlier that $\angle 3 = \angle 5$. We can replace $\angle 5$ with $\angle 3$ in the equation. This gives us the relation: $\angle 3 + \angle 6 = 180^\circ$ This relation holds true for any value of $\angle 3$. It is true if lines $l$ and $m$ are parallel.

Answer

The relation is that their sum is 180 degrees. This means they are supplementary angles. For example, if $\angle 3 = 50^\circ$, then $\angle 6 = 130^\circ$. This relation holds always if lines $l$ and $m$ are parallel.