Question

find the values of x and y in the picture

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. So the non - labeled angle in the top triangle and the non - labeled angle in the bottom triangle (the ones formed by the intersection of the two lines) are equal. Let's call this common angle \( z\). In the top triangle, using the triangle sum property (\(a + b + c=180^{\circ}\) for a triangle with angles \(a\), \(b\), \(c\)), we know that \(47^{\circ}+y^{\circ}+z = 180^{\circ}\). In the bottom triangle, \(x^{\circ}+z + 79^{\circ}=180^{\circ}\). But since the two lines are parallel (indicated by the arrow marks), we can also use the alternate - interior angles property. However, a simpler way is to use the fact that the two triangles have vertical angles (so the non - labeled angles are equal).

We know that \(x = 47\) (alternate - interior angles, because the lines are parallel and the transversal creates equal angles for the two non - adjacent non - straight angles of the two triangles).

Step2: Calculate \(y\)

Using the triangle sum property for the top triangle. The sum of angles in a triangle is \(180^{\circ}\). We know one angle is \(47^{\circ}\), and using the vertical angles (the non - labeled angle is equal to the angle in the other triangle. But since \(x = 47\) (from step 1, alternate - interior angles), and in the bottom triangle, if we consider the vertical angle of \(79^{\circ}\) (the non - labeled angle in the top triangle is equal to the non - labeled angle in the bottom triangle).

The sum of angles in the top triangle: \(47 + y+79=180\) (because the non - labeled angle in the top triangle is equal to the non - labeled angle in the bottom triangle which is part of the \(x + 79+\) non - labeled angle \(=180\) sum.

$$y=180-(47 + 79)$$

$$y=180 - 126$$

$$y = 54$$

Answer:

\(x = 47\), \(y = 54\)