Find the relationship between the angles and use the corresponding theorem/postulate.
m∠ 2=93^(∘)
m∠ 3=87^(∘)
m∠ 4=87^(∘)
Practice makes perfect
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 a, b, c, and d, and two transversals that are parallel to each other, f and e. We will mark the given angles on this diagram.
Let's calculate the measure of each of these angles one at a time.
Angle ∠ 2
We are given the measure of only one angle, and it is 93^(∘). Let's analyze the above diagram and try to find the relationship between the 93^(∘) angle and ∠ 2.
These angles lie on the opposite sides of the transversal a. They are interior angles formed by the parallel lines f and e. These are called alternate interior angles. Thus, in order to find the measure of ∠ 2, we can use Alternate Interior Angles Theorem.
If two parallel lines are cut by a transversal,
then each pair of alternate interior angles
is congruent.
By the theorem, the angles are congruent and have the same measures. Therefore, the measure of ∠ 2 is 93^(∘).
Angle ∠ 3
We need to look closely at the diagram and try to find the relationship between ∠ 3 and either ∠ 2 or the 93^(∘) 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 ∠ 3. However, the adjacent angle to ∠ 3 (let's call it ∠ 5) is a corresponding angle to ∠ 2. Using Corresponding Angles Postulate, we can find its measure and, perhaps, it will help us to find the measure of ∠ 3 later. Let's recall the mentioned postulate.
If two parallel lines are cut by a transversal,
then each pair of corresponding angles
is congruent.
We can conclude that ∠ 2 and ∠ 5 are congruent and have the same measures. Therefore, the measure of ∠ 5 is also 93^(∘). Now, let's consider angles ∠ 5 and ∠ 3. These are supplementary angles, so the sum of their measures is 180^(∘).
m∠ 5+m∠ 3=180^(∘)
Since we know the measure of ∠ 5, let's substitute m∠ 5 with 93^(∘) and solve the equation for m∠ 3.
93^(∘)+m∠ 3=180^(∘) ⇒
m∠ 3=180^(∘)- 93^(∘)= 87^(∘)
The measure of angle ∠ 3 is 87^(∘).
Angle ∠ 4
Similarly, we need to find the relationship between ∠ 4 and one of the other angles.
Angle ∠ 4 and the 93^(∘) angle lie on the same side of the transversal e, but on the different sides of the parallel lines a and d. These are consecutive interior angles. Thus, let's use the Consecutive Interior Angles Theorem.
If two parallel lines are cut by a transversal,
then each pair of consecutive interior angles
is supplementary.
By the theorem, these angles are supplementary. This tells us that the sum of their measures is 180^(∘).
m∠ 4+93^(∘)=180^(∘) ⇒
m∠ 4=180^(∘)-93^(∘)= 87^(∘)
The measure of the angle ∠ 4 is 87^(∘).
