Question

1. find ( mangle cxy ).

( mangle cxy= )

Explanation:

Step1: Use the SAS (Side - Angle - Side) congruence criterion

Since \(CX = CE/2\), \(CY=CD/2\), and \(\angle C\) is common. By the SAS similarity criterion, \(\triangle CXY\sim\triangle CED\).

Step2: Use the property of similar triangles and linear pair

We know that in \(\triangle CED\), let's first find \(\angle C\). Using the angle - sum property of a triangle in \(\triangle CED\), \(\angle C+\angle E+\angle D = 180^{\circ}\). But we can also use the property of the exterior angle.
Another approach: Since \(CX = CE/2\), \(CY = CD/2\) (by the mid - segment property in a triangle, \(XY\parallel ED\)).
The sum of an interior angle and its corresponding exterior angle is \(180^{\circ}\). If we consider the line \(CX\) and the transversal \(XY\) (with \(XY\parallel ED\)).
We know that \(\angle CXY\) and the angle adjacent to it (let's call it \(\angle A\)) form a linear pair.
Since \(XY\parallel ED\), we can use the property that \(\angle CXY\) and \(\angle XYE\) (where \(\angle XYE\) is related to the angles in the triangle).
By the mid - segment theorem (a special case of similar triangles where the ratio of sides is \(1:2\)).
We know that \(\angle CXY=180^{\circ}- 95^{\circ}\) (because of the parallel lines \(XY\parallel ED\) and the consecutive interior angles).
\(m\angle CXY = 180^{\circ}-95^{\circ}\)
\(m\angle CXY=85^{\circ}\)

Answer:

\(85^{\circ}\)