Question

if ( mangle x = 34^{circ}) and ( mangle z = 26^{circ}), what is ( mangle y)?
( mangle y = 26^{circ})
( mangle y = 34^{circ})
( mangle y = 60^{circ})
( mangle y = 120^{circ})

Explanation:

Step1: Use the property of exterior angles

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Calculate \(m\angle Y\)

Let's assume the exterior angle at \(Z\) (formed by the parallel lines) is equal to \(m\angle X + m\angle Y\). But if we consider the triangle \(XYZ\) (assuming the figure is a triangle - like structure with some parallel line properties for exterior angle consideration), using the angle - sum property of a triangle is wrong. Wait, actually, if we consider the property that the sum of angles in a triangle is \(180^{\circ}\) is not directly applicable here. Wait, no, using the exterior angle theorem (if we assume the two lines are parallel and we have a transversal - like situation for the exterior angle). Wait, another approach: Let's assume we use the formula \(m\angle Y=180^{\circ}-(m\angle X + m\angle Z)\) is wrong. Wait, no, actually, if we consider the fact that if we assume the two lines are parallel (from the figure's structure, the vertical lines), then the angle adjacent to \(Z\) (exterior) is \(m\angle X + m\angle Y\). But if we consider the triangle - like figure (even if it's a non - standard triangle due to the parallel lines), using the property that the sum of angles in a triangle - like structure (with the help of parallel line's alternate interior angles etc. to form a closed figure) is not straightforward. Wait, no, actually, using the formula for the sum of angles in a triangle: No, wait, no. Wait, using the exterior angle theorem: If we assume that the angle adjacent to \(Z\) (exterior) is equal to \(m\angle X + m\angle Y\). But if we consider that the two vertical lines are parallel, then the exterior angle at \(Z\) (let's call it \(E\)) is \(m\angle X + m\angle Y\). But if we assume that \(E = 180^{\circ}-m\angle Z\) (linear pair). So \(180^{\circ}-m\angle Z=m\angle X + m\angle Y\). Substituting \(m\angle X = 34^{\circ}\) and \(m\angle Z = 26^{\circ}\), we get \(180 - 26=34 + m\angle Y\). So \(154 = 34 + m\angle Y\), then \(m\angle Y=120^{\circ}\)

Answer:

\(m\angle Y = 120^{\circ}\)