Question

find the values of x and y in the picture

Explanation:

Step1: Use the property of vertical angles

Vertical angles are equal. So the non - \(41^{\circ}\), non - \(y^{\circ}\) angle in the top triangle and the non - \(x^{\circ}\) angle in the bottom triangle (the angle adjacent to \(77^{\circ}\)) are equal. Let's first find the third angle in the top triangle. The sum of angles in a triangle is \(180^{\circ}\). In the top triangle, if one angle is \(41^{\circ}\) and let the third angle (adjacent to \(77^{\circ}\)) be \(z\). Then \(z = 180-(41 + y)\). Also, in the bottom triangle, using the angle - sum property of a triangle (\(180^{\circ}\)), if one angle is \(x\) and the angle adjacent to \(77^{\circ}\) is \(180 - 77=103^{\circ}\). But since the two triangles have vertical angles (equal), we can also use the property of alternate - interior angles (because of the parallel lines indicated by the arrows).

Step2: Use the alternate - interior angles property

Since the lines are parallel (indicated by the arrows), \(x = 41^{\circ}\) (alternate - interior angles).

Step3: Use the angle - sum property of a triangle

For the top triangle, we know that the sum of angles in a triangle is \(180^{\circ}\). One angle is \(41^{\circ}\), and using the vertical angle (the angle opposite to \(77^{\circ}\) in the intersection of the two lines). The sum of angles in a triangle: \(41+y+(180 - 77)=180\). Simplify the equation: \(41+y + 103=180\). Then \(y+144 = 180\). Subtract 144 from both sides: \(y=180 - 144\).

Answer:

\(x = 41\), \(y = 36\)