The measures of the angles of a triangle are in the ratio 2 : 3 : 4. The simplified ratio of the measures of the exterior angles of the triangle is a : b : c. Find a + b + c.

18

Step-by-step 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.