Question

what are the values of x, y, and z? use theorems to justify each answer. by the corresponding angles theorem and the triangle angle - sum theorem, x = 40 (type a whole number.) by the angles theorem, y = (type a whole number.)

Explanation:

Step1: Identify vertical - angles

Vertical - angles are congruent. The angle vertical to the $60^{\circ}$ angle is also $60^{\circ}$.

Step2: Use triangle angle - sum theorem

In the triangle formed, we know one angle is $z$, another is the vertical - angle of $60^{\circ}$ ($60^{\circ}$) and the third is a $40^{\circ}$ angle (since $x = 40$). By the triangle angle - sum theorem, the sum of the interior angles of a triangle is $180^{\circ}$. So $z+60 + 40=180$, which gives $z=180-(60 + 40)=80$.

Step3: Identify corresponding angles

If we consider the parallel - line relationships, $y$ and the angle adjacent to the $60^{\circ}$ angle (which is $120^{\circ}$ since it forms a linear pair with $60^{\circ}$) are corresponding angles. So $y = 120$.

Answer:

$y = 120$, $z = 80$