Answer:
18
Explanation:
The measures of the angles of a triangle are in the ratio 2:3:4 means we have one interior angle measuring 2x, another measuring 3x, and the last one measuring 4x. We know the sum of the measures of the interior angles of a triangle is 180 degrees.
We need to solve the following equation to find first x, and then find the measurement of each interior angle of the triangle.
2x+3x+4x=180
(2+3+4)x=180
(9)x=180
x=180/9
x=20
So one interior angle measures 2x=2(20)=40.
Another measures 3x=3(20)=60.
The last measuring 4x=4(20)=80.
The exterior angles of this triangle therefore measure the following:
180-40=140
180-60=120
180-80=100
So the un-simplified ratio of the measures of the exterior angles is as follows:
140:120:100.
Let's simplify that.
It is easy to see each number is divisible by 10.
Let's reduce it by dividing each by 10:
14:12:10
Let's simplify more (as we want the most simplified ratio).
Each number is divisible by 2 since they are all even.
Let's reduce again but not by dividing by 2:
7:6:5
So the simplified ratio of the measures of the exterior angles of the triangle is a:b:c=7:6:5 and so a+b+c=7+6+5=18.
Answer is 18.