We want to show that z=x+y for the angles on the given diagram. Let's consider the diagram!
We can see that ∠ z is an exterior angle and ∠ x and ∠ y are two nonadjacent interior angles. This means that we want to show that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. We can mark the angles on the diagram.
Notice that ∠ w and ∠ z are supplementary angles. This means that they form a complete angle, so the sum of their measures is 180^(∘). By this property, we can write our first equation.
w+ z=180 (I)
Since we already found one equation involving ∠ w, we only need to find one more equation! We can use the rule for the interior angle measures of a triangle.
Interior Angle Measures of a Triangle
The sum of the measures of interior angles of a triangle is 180^(∘).
By this rule, the following equation is also true.
x+ y+w=180 (II)
We found two equations that contain w. We will use these two equations to write one equation the does not contain w to prove that z= x+ y. Let's isolate w in the first equation. This will give us an expression equal to w that we can substitute into the second equation. Let's do it!
w+ z=180 (I)
⇕
w = 180- z (I)
Finally, we can substitute the expression ( 180-z) for w in the second equation and solve for z.
The measure of z is equal to the sum of the measures of x and y. This means that the measure of an exterior angle of the triangle is equal to the sum of the measures of the nonadjacent interior angles. Notice that this statement is the same as the rule for the exterior angle measures of a triangle.
