When we extend one of the sides of a triangle, we form a new angle outside of the triangle. The angle formed is called an exterior angle. We know that one of the exterior angles of a triangle measures 120^(∘). Let's draw the triangle!
We want to find the measures of interior angles of the triangle. Notice that the exterior angle and the angle labeled as x^(∘) are supplementary. This means that they form a complete angle and that their measures sum up to 180^(∘).
120^(∘)+x^(∘)=180^(∘)Now, we can solve the equation for x.
The interior angle adjacent to the exterior angle measures 60^(∘). Next, we want to find the measures of the interior angles nonadjacent to the exterior angle. To do so, we can use the rule for the exterior angle measures of a triangle.
Exterior Angle Measures of a Triangle
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
We will now identify the two nonadjacent interior angles.
Notice that the nonadjacent interior angles are y^(∘) and z^(∘). By the rule, the following relation is true.
y^(∘)+ z^(∘)= 120^(∘)
Since we do not know the measures of the nonadjacent interior angles, nor do we know the relationship between them, the equation has two variables. When an equation has two unknown variables, we cannot solve it. This means that there is no unique solution and there are multiple triangles with one exterior angle that measures 120^(∘).
Possible Solutions
ccc
y=100 &and & x=20
y=60 &and & x=60
y=119 &and & x=1
In this case, we cannot determine all measures of interior angles of the triangle.
