Draw an auxiliary line where the arrows intersect. Then analyze the relationships between the angles and find their measures.
x=22
Practice makes perfect
We are given the following diagram and asked to find x. For the purpose of this solution, let's name the parallel lines t and p. We will also add a point B where the arrows intersect.
Because the given angles do not relate to each other, it is quite complicated to find the value of x. Thus, let's use the hint given in the exercise and draw an auxiliary line, a. The most useful line placement will be to draw the line parallel to t and p through point B. We will also name some useful angles on the diagram.
As we can see, the sum of m∠ 2 and m∠ 3 is equal to 72^(∘). We can use this to write the following equality.
m∠ 2+ m∠ 3=72^(∘)
Most likely we will need it later. Now, let's analyze the relationships between the angles and find their measures. Then we will be able to calculate the value of x.
Finding the Measure of ∠ 3
Let's take a look at the angle pair ∠ 3 and the given angle that measures 50^(∘).
These are corresponding angles. Thus, to find the measure of ∠ 3, we can use the Corresponding Angles Postulate.
If two parallel lines are cut by a transversal,
then each pair of corresponding angles
is congruent.
According to this postulate, the angles are congruent. This means that their measures are the same. Therefore, we conclude that m∠ 3=50^(∘).
Finding the Measure of ∠ 2
In the beginning of the solution, we wrote the equation to relate the measures of ∠ 2 and ∠ 3.
m∠ 2+ m∠ 3=72^(∘)
Now that we know m∠ 3, we can substitute it in the equation and solve it for m∠ 2.
Finally, we can analyze ∠ 2 and the angle that measures x^(∘).
As we can see, they are corresponding angles. Again, by the Corresponding Angles Postulate we can conclude that ∠ 2 and the angle that measures x^(∘) are congruent. This means that their measures are equal. Earlier we found that m∠ 2=22^(∘). Therefore, the value of x is 22.
