Question

1. find the measure of the exterior angle.

1. \\( \triangle a b c \\) is equiangular, \\( m \angle a = ( 6 x + 18 ) ^ { \circ } \\), and \\( m \\) for \\( x \\) and \\( y \\).
2. the figure shows three exterior angles of \\( \triangle a b c \\). show that \\( m \angle 1 + m \angle 2 + m \angle 3 = 360 ^ { \circ } \\).
3. find the values of \\( x \\) and \\( y \\).

Explanation:

Step1: Find the third angle of the triangle with \(68^{\circ}\) and \(x^{\circ}\)

The sum of angles in a triangle is \(180^{\circ}\). For the triangle with \(68^{\circ}\), the adjacent angle to \(85^{\circ}\) is \(180 - 85=95^{\circ}\). Then for the triangle with \(68^{\circ}\), \(x^{\circ}\) and \(95^{\circ}\), we use the angle - sum formula \(x + 68+95 = 180\). Simplify to \(x+163 = 180\).

Step2: Solve for \(x\)

Subtract \(163\) from both sides of the equation \(x+163 = 180\). So \(x=180 - 163=17\).

Step3: Find the third angle of the triangle with \(74^{\circ}\)

The sum of angles in a triangle is \(180^{\circ}\). The adjacent angle to \(85^{\circ}\) in the other large - triangle is \(180 - 85 = 95^{\circ}\). For the triangle with \(74^{\circ}\), \(y^{\circ}\) and \(95^{\circ}\), we use the angle - sum formula \(y+74 + 95=180\). Simplify to \(y + 169=180\).

Step4: Solve for \(y\)

Subtract \(169\) from both sides of the equation \(y + 169=180\). So \(y=180-169 = 11\).

Answer:

\(x = 17\), \(y = 11\)