Exercise 21 Page 508

To better understand how we should solve this, let's rotate the figure like below and extend a few of the rays.

Now we see that 5x and (14x-10)^(∘) are two angles that are between two parallel lines. They are also on the same side of the transversal that cuts those lines and therefore, we can classify them as consecutive interior angles. According to the Consecutive Interior Angles Theorem, such angles are supplementary which means we can write the equation 5x+14x-10=180 ⇔ 19x-10=180To find the second equation for our system of linear equations, we zoom in on the rotated figure, extend another one of the rays, and name one of the angles as 1.

From the figure, we see that ∠ 1 and (14x-10) are vertical angles. Therefore, by the Vertical Angles Congruence Theorem, they are congruent: m∠ 1=(14x-10)^(∘). We can also tell that ∠ 1 and 2y are consecutive interior angles which means they are supplementary: 14x-10+2y=180 Combining the two equation we create a system of equations which we can solve by Method of Substitution.

Having solved the first equation for x we can substitute this value in the second equation and solve for y.