Question

given that ( mangle a = 96^circ ) and ( mangle b = 68^circ ), find ( mangle x ) and ( mangle y ).
( mangle x=)
( mangle y=)
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Explanation:

Step1: Find \(m\angle x\)

Use the property that the sum of angles around a point is \(360^{\circ}\). Also, vertical - angle and linear - pair relationships.
Another way: The sum of the measures of angles forming a linear pair is \(180^{\circ}\). For the angle adjacent to \(\angle a\) (let's call it \(\angle c\)), \(m\angle c=180 - m\angle a\). Since \(m\angle a = 96^{\circ}\), \(m\angle c = 180-96=84^{\circ}\).
The sum of the measures of angles forming a linear pair is \(180^{\circ}\). For the angle adjacent to \(\angle b\) (let's call it \(\angle d\)), \(m\angle d=180 - m\angle b\). Since \(m\angle b = 68^{\circ}\), \(m\angle d = 180 - 68=112^{\circ}\).
Now, using the angle - sum property of a quadrilateral (the sum of interior angles of a quadrilateral formed by the intersecting lines is \(360^{\circ}\)) or the property of the sum of angles around a point of intersection of two lines.
\(m\angle x=m\angle c + m\angle d\) (by the exterior - angle property of a triangle formed by the intersecting lines). \(m\angle x=84 + 112=196^{\circ}\) (This is wrong. Let's use the correct property: The sum of angles around a point of intersection of two lines. The sum of angles around a point is \(360^{\circ}\). Also, \(\angle x\) and the sum of the two non - adjacent angles (\(m\angle a\) and \(m\angle b\)): \(m\angle x=m\angle a + m\angle b\) (by the exterior - angle property of a triangle formed by the intersecting lines). \(m\angle x=96+68 = 164^{\circ}\)

Step2: Find \(m\angle y\)

Since \(\angle y\) and the angle adjacent to \(m\angle b\) (the angle we called \(m\angle d = 112^{\circ}\)) are vertical angles. Or using the property that \(\angle y\) and \(m\angle b\) are supplementary to the same angle.
\(m\angle y=180 - m\angle b\) (because \(\angle y\) and \(\angle b\) are supplementary to the same angle formed by the intersecting lines). \(m\angle y = 180-68=112^{\circ}\) (Wrong. Let's use the correct property: \(\angle y\) and the angle adjacent to \(m\angle a\) (the angle we called \(m\angle c = 84^{\circ}\)) are vertical angles. So \(m\angle y=84^{\circ}\)

Answer:

\(m\angle x = 164^{\circ}\), \(m\angle y=84^{\circ}\)