Question

1. 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.

Explanation:

Step1: Find the interior - angle measures

The sum of the interior angles of a triangle is 180°. Let the interior angles be \(2x\), \(3x\), and \(4x\). Then \(2x + 3x+4x=180^{\circ}\), so \(9x = 180^{\circ}\), and \(x = 20^{\circ}\). The interior angles are \(2x=40^{\circ}\), \(3x = 60^{\circ}\), and \(4x=80^{\circ}\).

Step2: Find the exterior - angle measures

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles.
The exterior angles corresponding to \(40^{\circ}\), \(60^{\circ}\), and \(80^{\circ}\) are \(140^{\circ}\), \(120^{\circ}\), and \(100^{\circ}\) respectively.

Step3: Find the ratio \(a:b:c\)

The ratio of the exterior angles \(a:b:c=140:120:100 = 7:6:5\).

Answer:

\(7:6:5\)