We also see that the 110^(∘) angle is exterior angle to the triangle's vertex angle. By the Exterior Angles Theorem, the measure of the exterior angle equals the sum of the measures of the nonadjacent interior angles. With this information we can write and solve an equation for x.
Since the two lines cut by the transversal are parallel, the consecutive interior angles are supplementary. Additionally, two angles that form a straight angle must also be supplementary. With this, we can write two equations.
(5x+2^(∘))+63^(∘) = 180^(∘)
(3y-12^(∘))+63^(∘) = 180^(∘)
Let's solve for x in the first equation.
To find y, we will first have to find the vertex angle and the length of the sides that contain x-expressions. Since we know that the base angles are both 40^(∘), we can use the Interior Angles Theorem to find the measure of the vertex angle.
θ +40^(∘) +40^(∘)=180^(∘) ⇔ θ =100^(∘)
We can use either expression and that x= 5.5 to find the missing side lengths.
c
3 x+13=5 x+2
3( 5.5)+13=5( 5.5)+2
29.5=29.5
Now that we know the vertex angle and the side lengths, we can calculate the side labeled y by using the Law of Sines.
