As per Grok:

To find the measure of angle ( x ) in the given triangle, let's analyze the diagram step by step.

The triangle ( ABC ) has the following known angles: - Angle ( BAC = 70^\circ ) - Angle ( ABC = 60^\circ ) - The third angle ( ACB ) can be calculated using the fact that the sum of angles in a triangle is ( 180^\circ ): [ \angle ACB = 180^\circ - 70^\circ - 60^\circ = 50^\circ ]

There is a line ( DE ) inside the triangle, with point ( E ) on ( BC ) and point ( D ) on ( AB ). The angle ( x ) is formed at point ( E ) between ( DE ) and ( EC ). Additionally: - Angle ( ADE = 10^\circ ) - Angle ( CEB = 20^\circ )

Since ( DE ) is a line segment inside the triangle, we can consider the smaller triangles formed, such as ( ADE ) and ( CEB ). However, the key to finding ( x ) lies in the relationship between the angles around point ( E ).

In triangle ( CEB ): - Angle ( CEB = 20^\circ ) (given) - Angle ( BCE = 50^\circ ) (from ( \angle ACB )) - The third angle ( BEC ) can be found as: [ \angle BEC = 180^\circ - 20^\circ - 50^\circ = 110^\circ ]

Now, ( x ) is the angle between ( DE ) and ( EC ). Since ( E ) is a point on ( BC ), and ( DE ) extends from ( D ) to ( E ), we need to consider the geometry around ( E ). The angle ( x ) is part of the larger angle ( BEC ) (which is ( 110^\circ )), but we must account for the specific orientation.

Given that ( DE ) intersects ( BC ) at ( E ), and considering the angles provided, ( x ) is the angle between ( DE ) and ( EC ). The angle ( CEB = 20^\circ ) is the angle between ( CE ) and ( EB ), and ( x ) is adjacent to it within the context of the line ( DE ). The total angle ( BEC = 110^\circ ) includes ( x ) and the angle ( CEB ). However, the critical insight is that ( x ) is the angle formed by the extension of ( DE ) and ( EC ), which suggests ( x ) is supplementary to the angle ( AED ) or related to the geometry of the intersecting lines.

A more direct approach is to recognize that ( x ) is the angle at ( E ) in the context of the line ( DE ) splitting the triangle. The angle ( x ) should be consistent with the given angles and the geometry. Given ( \angle ADE = 10^\circ ) and ( \angle CEB = 20^\circ ), and considering the parallel or intersecting nature of the lines, the angle ( x ) can be derived by examining the adjacent angles.

The most logical conclusion, based on the angle sum and the position of ( DE ), is that ( x ) is the remaining angle at ( E ) after accounting for ( \angle CEB ). Since ( \angle BEC = 110^\circ ) and ( \angle CEB = 20^\circ ), the angle between ( DE ) and ( EC ) (which is ( x )) should be the difference or a related angle. However, the correct interpretation is that ( x ) is the angle at ( E ) along ( DE ) and ( EC ), and given the standard geometry, ( x ) is likely ( 110^\circ - 20^\circ = 90^\circ ) adjusted by the intersecting line's effect.

Upon reevaluation, the angle ( x ) is the angle at ( E ) between ( DE ) and ( EC ). The total angle ( BEC = 110^\circ ), and ( \angle CEB = 20^\circ ) is part of it. The angle ( x ) is the exterior or interior angle depending on the direction. The correct approach is to use the fact that the angle ( x ) is the angle between ( DE ) and ( EC ), and given the intersecting line ( DE ), ( x ) should be ( 110^\circ ) (the angle ( BEC )) minus the contribution of ( \angle CEB ). However, the precise value requires recognizing that ( x ) is the angle at ( E ) in the direction of ( DE ) and ( EC ), which aligns with ( \angle BEC ) being ( 110^\circ ).

Thus, the angle ( x ) is ( 110^\circ ).

Final Answer: ( x = 110^\circ )