McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
2. Angles and Parallel Lines
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Exercise 7 Page 183

We are told that the guard rail is parallel to the surface of the roadway and the vertical supports are parallel to each other. Using this information, we can draw four parallel lines and and two transversals that are parallel to each other, and We will mark the given angles on this diagram.

Let's calculate the measure of each of these angles one at a time.

Angle

We are given the measure of only one angle, and it is Let's analyze the above diagram and try to find the relationship between the angle and

These angles lie on the opposite sides of the transversal They are interior angles formed by the parallel lines and These are called alternate interior angles. Thus, in order to find the measure of we can use Alternate Interior Angles Theorem.
By the theorem, the angles are congruent and have the same measures. Therefore, the measure of is

Angle

We need to look closely at the diagram and try to find the relationship between and either or the angle.

Unfortunately, the relationship with any of these angles will not allow us to use one of the known theorems/postulates and find the measure of However, the adjacent angle to (let's call it is a corresponding angle to Using Corresponding Angles Postulate, we can find its measure and, perhaps, it will help us to find the measure of later. Let's recall the mentioned postulate.
We can conclude that and are congruent and have the same measures. Therefore, the measure of is also Now, let's consider angles and These are supplementary angles, so the sum of their measures is
Since we know the measure of let's substitute with and solve the equation for
The measure of angle is

Angle

Similarly, we need to find the relationship between and one of the other angles.

Angle and the angle lie on the same side of the transversal but on the different sides of the parallel lines and These are consecutive interior angles. Thus, let's use the Consecutive Interior Angles Theorem.
By the theorem, these angles are supplementary. This tells us that the sum of their measures is
The measure of the angle is