Find angle 1.

Solution:

We are given a diagram with intersecting lines and angles. We need to find the measure of angle 1.

Observe the angle marked 110°. This is an exterior angle to a triangle formed by the intersecting lines. The interior angles of this triangle are angle 1, another angle that is vertically opposite to an angle marked 52°, and an angle adjacent to the 110° angle on a straight line.

Let's denote the vertex where angles 52° and 110° are located as V.

The angle adjacent to 110° on the straight line is \( 180° - 110° = 70° \). This is an interior angle of the triangle.

The angle marked 52° is given. The angle vertically opposite to it is also 52°.

The sum of angles in a triangle is 180°.

In the triangle, the angles are angle 1, 52°, and 70°.

So, \( \angle 1 + 52° + 70° = 180° \).

\( \angle 1 + 122° = 180° \)

\( \angle 1 = 180° - 122° \)

\( \angle 1 = 58° \)

Alternatively, consider the transversal line intersecting two parallel lines (implied by the notation of the top and bottom horizontal lines usually representing parallel lines in such diagrams). The angle 70° and the angle adjacent to it on the straight line (which is 110°) are related to the transversal. The angle 70° is an alternate interior angle to the angle formed by the transversal and the bottom line, on the right side of the transversal. The angle 110° is an exterior angle.

Let's assume the top and bottom horizontal lines are parallel.

The angle given as 70° is an alternate interior angle to the angle formed by the transversal and the bottom line, to the right of the point where the transversal intersects the bottom line.

Consider the transversal that forms angle 1. The angle 52° and the angle adjacent to it on the straight line are within a triangle. The sum of angles on a straight line is 180°.

The angle supplementary to 110° is \( 180° - 110° = 70° \). This angle and the angle marked 52° are interior angles of the triangle. Angle 1 is the third interior angle.

Therefore, in the triangle, the sum of angles is \( \angle 1 + 52° + 70° = 180° \).

\( \angle 1 = 180° - (52° + 70°) \)

\( \angle 1 = 180° - 122° \)

\( \angle 1 = 58° \)

Final check: If angle 1 is 58°, then the angles of the triangle are 58°, 52°, and 70°. The sum is \( 58 + 52 + 70 = 110 + 70 = 180° \). This is correct.

Ответ: 58°.