Glencoe Math: Course 3, Volume 2
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Glencoe Math: Course 3, Volume 2 View details
1. Lines
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Exercise 10 Page 376

We are given the following diagram.

A parallelogram whose long sides are AB and DC. Ray AF passes through point F, which is collinear with A and B but not on side AB. Ray DE passes through point E, which is collinear with D and C but not on side DC.

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

The parallelogram with lengthen sides

Since 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 and To do so, we will start by marking an extra angle in the diagram.

The extra angle

Now let's analyze the relationship between and

The alternate interior angles
Notice that and 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 and
The supplementary angles
We can see that and form a straight line. This means that these angles are supplementary and the sum of their measures is
Finally, to get the relationship of and we can substitute for into this equation.
We found that the sum of and is meaning they are supplementary angles.