Question

1. find the values of x and y.

Explanation:

Step1: Find the value of \(x\)

Use the property that the sum of angles in a triangle is \(180^{\circ}\). For the left - hand triangle, we know one angle is \(68^{\circ}\), and we can find the third angle (let's call it \(a\)) in the non - overlapping part of the two triangles. The vertical angle to \(85^{\circ}\) is also \(85^{\circ}\). In the small triangle with \(50^{\circ}\), the third angle (which is equal to \(a\)) is \(180^{\circ}-(85^{\circ} + 50^{\circ})=45^{\circ}\). Then for the left - hand triangle, \(x=180^{\circ}-(68^{\circ}+45^{\circ}) = 67^{\circ}\).

Step2: Find the value of \(y\)

For the right - hand triangle, first find the non - \(74^{\circ}\) non - \(y\) angle. The vertical angle to \(85^{\circ}\) is \(85^{\circ}\). Using the angle sum property of a triangle, the non - \(74^{\circ}\) non - \(y\) angle is \(180^{\circ}-(85^{\circ}+74^{\circ}) = 21^{\circ}\). Then \(y = 180^{\circ}-21^{\circ}=159^{\circ}\).

Answer:

\(x = 67^{\circ}\), \(y=159^{\circ}\)