Draw an auxiliary line where the arrows intersect. Then find the measures of two angles that create the angle of measure x^(∘).
x=130
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 l and m. We will also add a point A 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 l and m through point A. We will also name some useful angles on the diagram.
The sum of m∠ 2 and m∠ 3 is equal to x^(∘). We can use this to write an equation.
x^(∘)=m∠ 2+ m∠ 3
Thereby, to find x, we need to find the measures of these angles. Let's do it!
Finding the measure of ∠ 2
In order to find the measure of ∠ 2, we should identify its relationship with one of the given angles. For example, ∠ 1 and ∠ 2 are consecutive interior angles. However, we do not know the measure of ∠ 1 yet. Is it possible to find it? Fortunately, yes!
Let's consider ∠ 1 and the given angle that measures 105^(∘).
These are vertical angles. From the Vertical Angles Theorem, we know that vertical angles are congruent. Therefore, the measure of ∠ 1 is also 105^(∘). Now, we are able to find the measure of ∠ 2. Since ∠ 1 and ∠ 2 are consecutive interior angles, we can use Consecutive Interior Angles Theorem.
If two parallel lines are cut by a transversal,
then each pair of consecutive interior angles
is supplementary.
According to this theorem, ∠ 1 and ∠ 2 are supplementary angles. This means that the sum of their measures is 180^(∘).
m∠ 1+ m∠ 2=180^(∘)
We already know m∠ 1, so we can substitute it in the equation and solve it for m∠ 2.
Now, we are going to find the measure of ∠ 3. Let's try to find its relationship with one of the other angles.
As we can see, ∠ 3 and ∠ 4 are also consecutive interior angles. Thus, we can find m∠ 3 the same way we found m∠ 2. By the Consecutive Interior Angles Theorem, ∠ 3 and ∠ 4 are supplementary angles and will add to 180^(∘).
m∠ 3+m∠ 4=180^(∘)
Again, we do not know the measure of ∠ 4. However, from the diagram, we can see that ∠ 4 and the given angle that measures 125^(∘) are vertical angles.
Therefore, they are congruent and their measures are the same. We conclude that m∠ 4= 125^(∘). Let's substitute this piece of information into the above equation.
m∠ 3+ 125^(∘)=180^(∘)
Subtracting 125^(∘) from both sides of the equation, we can calculate m∠ 3.
m∠ 3=180^(∘)-125^(∘)=55^(∘)
The measure of the angle ∠ 3 is 55^(∘).
Value of x
In the beginning of the solution, we wrote an equation to relate x to the measures of ∠2 and ∠3.
x^(∘)= m∠ 2+ m∠ 3
Now that we know the measures of both ∠ 2 and ∠ 3, we can substitute them into our equation.
lm∠ 2= 75^(∘) m∠ 3= 55^(∘) } ⇒ 75^(∘)+ 55^(∘)=130^(∘)
The value of x is 130.
