Exercise 10 Page 376

We are given the following diagram.

We are asked to make a conjecture about the relationship of $\angle 1$ and $\angle 2.$ Let's start by analyzing the diagram.

Since $ABCD$ is a parallelogram, its opposite sides are parallel. Therefore, if we lengthen three sides of the parallelogram, we get a line that intersects two parallel lines. When two parallel lines are cut by a transversal, special angle relationships exist. Let's find the relationship of $\angle 1$ and $\angle 2!$ To do so, we will start by marking an extra angle in the diagram.

Now let's analyze the relationship between $\angle 1$ and $\angle 3.$

Notice that $\angle 1$ and $\angle 3$ are interior angles that lie on opposite sides of the transversal. This means that these angles are alternate interior angles. Since the two horizontal lines are parallel, the measures of these angles are equal. Now let's take a closer look at $\angle 2$ and $\angle 3.$ We can see that $\angle 2$ and $\angle 3$ form a straight line. This means that these angles are supplementary and the sum of their measures is $180^{\circ }.$ Finally, to get the relationship of $\angle 1$ and $\angle 2,$ we can substitute $m\angle 1$ for $m\angle 3$ into this equation. We found that the sum of $m\angle 1$ and $m\angle 2$ is $180^{\circ },$ meaning they are supplementary angles.