I have this problem where Valentin draws a zig-zag line inside a rectangle with given angles: 10°, 14°, 33°, and 26°. I need to find the measure of angle φ. The diagram shows a rectangle with a zig-zag line creating several angles, and φ is one of them.

First, I should understand how these angles relate to each other. Since it's a rectangle, all angles are right angles, so 90 degrees. The zig-zag line is made up of segments that create these smaller angles with the sides of the rectangle.

Looking at the diagram, there are several triangles formed by the zig-zag line and the sides of the rectangle. The angles given are probably the acute angles between the zig-zag segments and the rectangle's sides.

Let me sketch this out mentally. The rectangle has horizontal and vertical sides. The zig-zag line goes from one side to another, changing direction at points.

From the angles, I see that at each vertex where the zig-zag meets the side, there's an angle given, like 10°, 14°, 33°, and 26°. Also, φ is somewhere in the middle.

In the diagram, it looks like there are four points where the zig-zag intersects the sides, creating these angles.

Perhaps the sum of these angles relates to the rectangle's angles.

Since the rectangle has 90-degree angles, and the zig-zag creates deviations, maybe the sum of the angles on one side should be 90 degrees.

For example, consider the top side of the rectangle. There might be multiple angles adding up to 90 degrees because of the zig-zag.

Similarly for other sides.

Let me label the points. Suppose the rectangle has corners A, B, C, D, with A top-left, B top-right, C bottom-right, D bottom-left.

The zig-zag line starts from the left side, goes to the right, but at an angle, then to the bottom, then to the left, then to the top, or something like that. I need to interpret the diagram.

From the angles given: 10° at the top, I think, 14° next, then φ, then 33° and 26° at the bottom.

The diagram shows angles at the intersections: 10° at the top-left part, 14° below that, then φ in the middle, then 33° and 26° at the bottom.

Perhaps it's better to think in terms of the angles with the horizontal and vertical sides.

Each segment of the zig-zag makes an angle with the side it meets.

For instance, the first segment from the left side might make a 10° angle with the top side, but since it's inside, it's probably with the vertical side.

I think I should consider the angles between the zig-zag segments and the sides.

Let me denote the zig-zag points. Let’s say the zig-zag line starts at the left side at point P, then goes to point Q on the top side, then to R on the right side? No, the angles are given at the sides, not at the corners.

From the diagram, it seems like the zig-zag has vertices on the sides of the rectangle, not at corners.

So, let's assume there are four points on the sides where the zig-zag changes direction.

For example: point A on left side, point B on top side, point C on right side, point D on bottom side, but that might not be right because the angles are given at different positions.

Looking at the angles: 10° is near the top-left, 14° is below it, φ is in the center, 33° and 26° are near the bottom.

Perhaps it's like this: the zig-zag line starts from the left side, makes a 10° angle with the top, but since it's on the left side, it might be with the vertical.

I think I need to think about the angles that the segments make with the horizontal or vertical.

Since the rectangle has horizontal and vertical sides, the angles given are probably the acute angles between the zig-zag segments and these sides.

For example, at the top side, there is an angle of 10°, which might be between the top side and the first segment of the zig-zag.

Similarly, 14° might be with the next segment.

But the segments are not necessarily on the same side; they are on different sides.

Let me describe the diagram as I see it.

From the image: there is a rectangle. On the left side, there is a point where the zig-zag starts or something. Actually, the zig-zag line is inside the rectangle, so it has multiple segments.

The angles 10°, 14°, 33°, 26° are labeled at the points where the zig-zag meets the boundary of the rectangle.

So, there are four points on the boundary: one on the top side, one on the top but closer to the left? Let's list the angles and their positions.

From the diagram: at the top-left, there is a 10° angle. This is probably the angle between the top side and the first segment of the zig-zag.

Similarly, below that, on the left side, there is a 14° angle, which might be between the left side and the next segment.

Then, in the middle, φ is an angle inside, not on the boundary.

Then on the bottom, there is a 33° angle and a 26° angle.

The 33° and 26° are on the bottom side, I think.

Let's assume the boundary points are: point P on the top side, where the angle is 10°. Point Q on the left side, where the angle is 14°. Point R on the bottom side, where the angle is 33°, and point S on the bottom side or right side? 26° is also on the bottom, I think.

In the diagram, 33° and 26° are both on the bottom side, with 33° on the left and 26° on the right or something.

φ is not on the boundary; it's an internal angle at the vertex of the zig-zag.

The zig-zag has a vertex inside the rectangle where the direction changes, and φ is the angle at that vertex.

Similarly, the other angles are at the boundary intersections.

So, let's define the points.

Let me say the rectangle has sides: top, right, bottom, left.

The zig-zag line enters from the left side at point A, then goes to an internal point B, then to a point on the top side C, then to another internal point D, then to a point on the right side E, but the angles given are 10° and 14° on the left/top, and 33° and 26° on the bottom, so it's not symmetric.

Perhaps it's a single zig-zag with multiple turns.

From the diagram, it looks like there is a point on the left side, then the zig-zag goes to the top, but at an angle, then from there to the right, but not to the corner, to a point on the top, then down to a point on the right, but the angles are given at the turns.

I think I found a better way: in such problems, the sum of the angles on one side of the rectangle must be 90 degrees because the zig-zag line divides the rectangle, and the angles at the boundary add up.

For example, on the top side, the angles between the top side and the zig-zag segments should add up to 90 degrees.

Similarly for other sides.

Let's consider the top side of the rectangle.

From the diagram, on the top side, there is one angle given: 10°. But there might be more.

At the left end of the top side, there is a 10° angle, which is the angle between the top side and the first segment of the zig-zag.

Then, at the right end, there might be another angle, but it's not given, so perhaps it's 90 degrees minus something.

Similarly, on the left side, there is a 14° angle at the top, and at the bottom, there is 26° or 33°, but 26° and 33° are on the bottom.

Let's list all the boundary angles:

• On the top side, at the left, angle 10° (between top and the zig-zag segment)

• On the left side, at the top, angle 14°? No, the 14° is below the 10°, so on the left side, there is a point where the angle is 14°, which is between the left side and the zig-zag segment.

Similarly, on the bottom side, there is a 33° angle and a 26° angle, both on the bottom side, I think.

In the diagram, for the bottom side, there are two angles: 33° on the left and 26° on the right.

And for the top side, only one angle: 10° on the left.

But at the right end of the top side, there must be an angle as well, but it's not labeled, so it must be part of the zig-zag.

Perhaps the zig-zag starts at the left side with angle 10° at the top, but the 10° is at the top-left corner area.

I think I need to assume the configuration.

Another idea: the angles given are the angles that the zig-zag segments make with the sides, and for the rectangle, the sum of the angles on each side must be 90 degrees.

For the top side: there is one angle given, 10°, but there might be another segment meeting the top side at the right end.

Similarly for other sides.

Let's count the number of segments.

From the diagram, it seems there are three segments: one from left to top, one from top to down, and one from down to right or something.

But there is φ in the middle, so there are multiple vertices.

Let's think of the zig-zag as having three turns, with four segments.

But the angles are given at the boundaries.

Perhaps the 10° and 14° are on the left side, 33° and 26° on the bottom, and φ is internal.

Let's describe the points:

• Point P on the left side: at P, the angle between the left side and the first zig-zag segment is 10°? But the 10° is labeled at the top, so it's confusing.

I think the 10° is the angle at the top side near the left corner.

Let's read the diagram: the 10° is at the top-left, so it's the angle between the top side and the segment from the left side to the first internal point.

Similarly, the 14° is on the left side, below the 10°, so it's the angle between the left side and the segment from the first internal point to the second, or something.

I think I have it: the zig-zag line has a vertex at the left side, but the angle is measured at the boundary.

Standard way: at each point where the zig-zag meets the boundary, there is an angle between the boundary and the zig-zag segment.

For the top side, at point C (on top), the angle is 10°.

For the left side, at point A (on left), the angle is 14°.

Then, there is an internal vertex B where the angle is φ.

Then, on the bottom side, at point D (on bottom), the angle is 33°, and at point E (on bottom), the angle is 26°.

But there are two points on the bottom side? In the diagram, 33° and 26° are both on the bottom side, with 33° on the left and 26° on the right.

Similarly, on the top side, only one point with 10°, but at the right end, there might be another point.

Perhaps for the top side, at the right end, there is a point with an angle, but it's not labeled, so it must be that the zig-zag ends or something.

Let's assume the zig-zag starts at the left side with point A, angle 14° (since 14° is on left), then goes to internal point B, then to top side point C, angle 10° at C, then from C to another point, but the next angle is φ at B, but B is internal.

The segments are: from A on left to B internal, from B to C on top, then from C to D on bottom? But C is on top, so from C on top to D on bottom, that would be a diagonal, but it's a zig-zag, so it should change direction.

From B internal, it goes to C on top, then from C, it should go to another point, but in the diagram, after C on top, it goes down to a point on the bottom, but there is no angle at C for the top side; the 10° is at C for the top side.

At point C on top, the angle 10° is between the top side and the segment from B to C.

Similarly, at point A on left, the angle 14° is between the left side and the segment from A to B.

Then at B internal, the angle φ is between the two segments: A to B and B to C.

Then from C, the zig-zag goes to a point on the bottom, say D, but at D, there is an angle, but in the diagram, after C, it goes to a point on the bottom with 33°, but 33° is on the bottom, and there is also 26° on the bottom, so there must be two points on the bottom.

Perhaps from C on top, it goes to an internal point E, then to D on bottom with 33°, then to F on bottom with 26°, but that seems messy.

Another possibility: the 33° and 26° are at the same point or something? No, they are separate.

Let's look at the diagram: the 33° is on the bottom-left, 26° on the bottom-right, and between them, there is the zig-zag with φ.

Also, on the top, 10° on the left, 14° on the left side, but 14° is on the left side, not on top.

I think I misread.

The 10° is on the top side, at the left.

The 14° is on the left side, below the top.

Then, the zig-zag goes to an internal point with φ, then to the bottom side with 33° and 26°.

But 33° and 26° are both on the bottom side, so there are two points on the bottom side.

Similarly, on the left side, there is one point with 14°, and on the top side, one point with 10°, but at the corners, there are the corners themselves.

For the left side, at the top, there is the corner, but the 14° is below it, so the angle at the top-left corner is 90 degrees, but the zig-zag is not at the corner; it's at a point on the side.

So, for the left side, at point A, the angle between the left side and the zig-zag segment is 14°. This means that the zig-zag segment is not horizontal or vertical; it makes 14° with the left side.

Similarly for others.

Then, the internal angle at B is φ.

Then on the bottom side, at point D, the angle is 33°, and at point E, the angle is 26°.

But for the bottom side, at the left end, there is point D with 33°, and at the right end, point E with 26°, and between them, the bottom side is straight, so the angles at D and E are with the zig-zag segments.

Now, for the right side, there are no given angles, so perhaps the zig-zag ends or goes to the right side.

In the diagram, after the bottom, it goes to the right, but no angle given.

Similarly, for the top side, at the right end, no angle given.

So, perhaps the zig-zag starts at the left side with angle 14°, goes to internal point B, then to top side with angle 10°, then from B to C on top, then from C it must go to the bottom or something.

From C on top, it goes to a point on the bottom, but there is no direct segment; there is the internal point B, but B is already used.

The segments are: from A on left to B internal, from B to C on top, then from C, it goes to another point, but in the diagram, after C, it goes down to the bottom, but there is no other internal point mentioned; the φ is at B, which is for the turn from A to B to C.

Then from C, it should go to a point on the bottom, say D, with angle at D.

But in the diagram, there is 33° on bottom-left and 26° on bottom-right, so D is one, and E is another.

But from C on top to D on bottom, that would be a diagonal, not a zig-zag, so it must change direction at another point.

Perhaps there is another internal point.

The diagram shows only one φ, so probably only one internal vertex.

Let's count the angles: there are four angles given: 10°, 14°, 33°, 26°, and one to find φ.

Also, for the rectangle, there are four corners with 90 degrees each.

At each point where the zig-zag meets the side, the angle between the side and the segment is given, and at the internal vertex, the turn angle is φ.

For the sides, the sum of the angles on each side must be 90 degrees because the sides are straight.

For example, on the left side: the left side is vertical. At the top, there is the top-left corner, which has 90 degrees, but the zig-zag is not at the corner; it's at a point on the side.

So, on the left side, there is a point A where the zig-zag meets, and the angle at A is 14°, which is the angle between the left side and the zig-zag segment.

Since the left side is vertical, the zig-zag segment makes 14° with the vertical.

Similarly, at the top, on the top side, at point C, the angle is 10°, so the zig-zag segment makes 10° with the horizontal top side.

At the bottom, at point D, 33° with the horizontal bottom side, and at point E, 26° with the horizontal.

But for the left side, at point A, the angle is 14° with the vertical, so the deviation from vertical.

Now, for the left side, the total angle from the top to bottom is 90 degrees. At the top, there is the top-left corner, which is 90 degrees, but the corner is a point, and the side has points on it.

The left side from top to bottom is 90 degrees. At the top, there is the corner, and at the bottom, the corner.

At point A on the left side, the angle between the left side and the zig-zag segment is 14°. This means that from the horizontal, the zig-zag segment has an angle.

Let's define the angles with the sides.

For the left side (vertical): at point A, the angle between the left side and the segment A to B is 14°. Since the left side is vertical, the segment A to B makes 14° with the vertical, so its slope is such that the angle with horizontal is 90° - 14° = 76°, but I'm not sure if that helps.

Similarly, for the top side (horizontal): at point C, the angle between the top side and the segment B to C is 10°, so the segment B to C makes 10° with the horizontal, so its angle with vertical is 80°, etc.

But at the internal point B, the angle φ is the angle between the two segments: A to B and B to C.

The direction change.

In terms of vectors, the turn angle.

But we have the bottom angles as well.

Also, for the bottom side, there are two points, D and E.

So, the zig-zag must have more segments.

From C on top, it goes to B, but B is already connected to A, so from B to C is one segment, from A to B is another, then from C, it must go to another point.

Perhaps the internal point B is only for the first turn, and there is another internal point for the second turn.

But the diagram only shows one φ, so probably only one internal vertex.

Let's look at the diagram again: in the image, there is a rectangle with a line that goes from the left side, angles down at 14°, then to the top at 10°, but with an internal point where φ is, then from there, it goes down to the bottom with 33° and 26°.

But 33° and 26° are both on the bottom, so it must be that from the internal point, it goes to one point on the bottom with 33°, then to another point with 26°, but that would require another vertex.

Perhaps the 33° and 26° are at the same point, but they are different angles, so no.

Another idea: the 33° and 26° are the angles at two different points on the bottom side, and between them, the bottom side is straight, so the angles at D and E are with their respective zig-zag segments.

But for the zig-zag, from the internal point B, it might go directly to a point on the bottom, but then only one point, but there are two angles, so two points.

I think I have it: the zig-zag line has two internal vertices or something.

Let's list the points in order.

Assume the zig-zag starts at point A on the left side with angle 14°, goes to internal point B, then to point C on the top side with angle 10°, then from C, it goes to another internal point, but there is only one φ, so perhaps from B, after C, it goes to another point.

The segment from B to C is from internal to top, so at C, the angle is 10° for the top side.

Then from C, the next segment is to a point on the bottom, say D, but at D, the angle is 33°, for example.

Then from D to another point, but on the bottom, there is also 26°, so perhaps to E on bottom with 26°, but then from D to E is along the bottom, but the zig-zag should be a line, so it must go to the right or something.

Perhaps from D, it goes to the right side or to another internal point.

But the diagram has φ only at B, so for the bottom, it might be that the segment from C to D is to the bottom, and at D, angle 33°, then from D to E on bottom with angle 26°, but that would be a straight line from D to E along the bottom, so the angle at D for the segment from C to D and D to E would be 180 degrees, but it's not labeled, and the 26° is at E for the bottom side.

I'm confused.

Let's think differently. Perhaps the 33° and 26° are for two different segments on the bottom.

But for the rectangle, on the bottom side, the sum of the angles from the left to the right must be 90 degrees.

Similarly for other sides.

For the bottom side: there are two points, D and E, with angles 33° and 26°.

At the left end, there is the bottom-left corner, and at the right end, the bottom-right corner.

The angle at D is 33°, which is between the bottom side and the zig-zag segment at D.

Similarly at E, 26° between bottom side and zig-zag segment at E.

Since the bottom side is horizontal, the segments at D and E make 33° and 26° with the horizontal.

The distance between D and E along the bottom is part of the side, but the total angle from the left corner to the right corner is 90 degrees.

At the left corner, the angle is 90 degrees between bottom and left side, but the zig-zag is not at the corner, so the contribution from the corners is zero for the zig-zag angles.

The points on the side divide the side into segments, and at each point, the angle with the segment determines how much the zig-zag turns.

For the bottom side, from left to right: bottom-left corner, then point D, then point E, then bottom-right corner.

At point D, the angle between the bottom side and the zig-zag segment is 33°. This means that the zig-zag segment at D is 33° from the horizontal.

Similarly at E, 26° from the horizontal.

The bottom side between D and E is straight, so the turn at D and E are separate.

For the zig-zag line, at D, it may be a vertex or not, but in this case, D and E are points where the zig-zag meets the boundary, so they are vertices of the zig-zag.

Similarly for A and C.

So, the zig-zag has vertices at A (on left), B (internal), C (on top), D (on bottom), E (on bottom), but E is on bottom, and from C to D to E, but D and E are both on bottom, so the segment from C to D is from top to bottom, then from D to E is along the bottom? That doesn't make sense for a zig-zag; it should change direction.

Unless from C to D is one segment, then from D to another point, but D is on bottom, so from D, it could go to the right or up, but to E on bottom, that would be along the bottom, so no turn, but the 26° is at E, so E is a different point.

Perhaps from D, it goes to the right side or to an internal point, but there is no other internal point.

I think the only way is that there is another internal point.

But the diagram has only one φ.

Perhaps for the bottom, the 33° and 26° are for the same point, but that can't be.

Another possibility: the 33° and 26° are the angles of the zig-zag segments with the bottom side, but at different points, and the zig-zag has a turn at an internal point with φ.

But in the diagram, between 33° and 26°, there is φ, so perhaps the internal point is for the turn between the segment to 33° and from 26°, but 26° is at a different point.

I think I found a key point: in the diagram, the 33° and 26° are on the bottom, but they are for the segments coming from above.

Let's assume that the zig-zag has five vertices: A on left, B internal, C on top, D on bottom with 33°, and E on bottom with 26°, but from D to E is not a segment of the zig-zag; the zig-zag segments are A to B, B to C, C to D, and then from D to where? If to E, it would be a segment, but on the bottom, so it must be that from C to D is to D on bottom, then from D to a point on the right or bottom-right, but E is on bottom, so it doesn't make sense.

Unless E is on the right side, but the angle 26° is labeled on the bottom in the diagram.

In the image, 26° is on the bottom side, at the right.

Similarly, 33° on bottom left.

So, there are two points on the bottom side.

For the zig-zag to have a turn, there must be a vertex on the bottom or internal.

But the only internal vertex has φ.

Perhaps the segment from C to D is from top to bottom, and at D, the angle is 33° for the segment from C to D and the bottom side.

Then from D, the next segment is to the right, to a point on the right side, but the 26° is on the bottom, not on the right.

26° is on the bottom, so it must be that at E, the angle is 26° for the bottom side, so E is another point on bottom.

So, the only way is that from C on top, it goes to D on bottom with segment CD, angle at D 33° with bottom.

Then from D, it goes to E on bottom with segment DE, but DE is along the bottom, so the angle at D for the zig-zag would be the angle between CD and DE. Since DE is along the bottom, and CD is coming down, the angle at D for the zig-zag is 180° - 33° or something, but it's not labeled, and at E, the angle 26° is between the bottom and the next segment, say to the right.

But then from E to a point on the right side.

And for the top side, at the right end, there might be a point, but not labeled.

But we have the internal point B with A to B and B to C.

So the zig-zag is: A (left) to B (internal), B to C (top), C to D (bottom), D to E (bottom), E to F (right), but D to E is along the bottom, so it's not a zig-zag segment; it's part of the side, so the zig-zag should not have a segment along the side; it should be inside.

So D to E should not be a segment; the points D and E are where the zig-zag meets the bottom, but the segment between them is not part of the zig-zag.

So for the bottom side, there are two points, D and E, where the zig-zag meets the bottom, so at each, there is a segment leaving the bottom.

At D, the segment from C to D arrives at the bottom, and at D, the angle is 33° between the bottom and the segment, but the segment is coming from above, so the angle is between the bottom and the direction of the segment.

Similarly at E, the segment from E to some point, with angle 26°.

But between D and E, on the bottom side, there is no zig-zag activity, so the segment from D to the next point must go to E or to another point.

If it goes to E, then D to E is a segment, but on the bottom, so it should be vertical or something, but the bottom is horizontal, so D to E horizontal would be along the side, which is not inside.

So the only possibility is that from D, the zig-zag goes to an internal point or to the right side.

But there is no other internal point, so it must go to the right side.

Similarly, at E, it might be that the segment to E is from above.

Let's assume that from C on top, the segment goes to D on bottom with angle 33° at D.

Then from D, the segment goes to the right side, say to F on right side, with an angle at F, but not given.

Then from F to E on bottom with angle 26° at E? That doesn't make sense.

E is on bottom, so if from F on right to E on bottom, that would be a segment, but E is on bottom, so at E, the angle is 26° for the bottom side.

But from D to F to E, but F is on right, so at F, there is an angle with the right side.

But no angles given for right side.

Moreover, the internal point B is for A to B to C, so C to D is another segment.

Then from D to F on right, then from F to E on bottom? E on bottom, but F on right, so from F to E is diagonal to bottom, but E is on bottom, so it could be, but then at E, the segment from F to E arrives, and the angle at E is 26° with the bottom.

But the bottom has points D and E, with D on left, E on right, so from D to E is not direct.

The segment from C to D is from top to bottom, so it's diagonal, then from D to F on right, also diagonal, then from F to E on bottom, which is from right to bottom, so it might be vertical or something.

But in the diagram, for the bottom, 33° at D and 26° at E, so D and E are different.

But with the internal point B only for the top part.

But we have φ at B for the turn.

For the bottom, there is no internal point, so the turn at the bottom must be at the boundary points.

At D, for example, the angle between the segment C to D and the bottom side is 33°, but that's the arrival angle, not the turn angle for the zig-zag.

The turn angle for the zig-zag at a vertex is the change in direction.

At D, if the segment comes from C to D, and then goes to F on right, the turn angle at D is the angle between the incoming and outgoing segments.

But at the boundary, the angle with the side is given, which relates to the direction.

I'm getting messy.

Let's look for a different approach.

I recall that in such zig-zag lines in a rectangle, the sum of the angles on one side must equal 90 degrees or something.

For example, on the top side: there is one point C with angle 10°. At the left end, the top-left corner has 90 degrees, but the zig-zag is not at the corner, so the angle from the left corner to point C: from the left corner, the side is vertical, but the top side is horizontal, so from the top-left corner, along the top side to C, the angle is 0 until C, then at C, the zig-zag segment makes 10° with the top side.

Similarly, at the right end of the top side, there might be another point or the corner.

But in this case, only one point on top, so from left corner to C, the side is straight, so the angle contribution is only at C, 10°, and from C to right corner, no angle, so the total for the top side is 10°, but it should be 90 degrees? No.

The total turn or something.

Perhaps for the rectangle, the sum of the acute angles at the boundary points must be such that the deviations compensate.

Another idea: the angles with the sides determine the slope of the segments.

Let's assume the segments.

Suppose the first segment from A to B: at A, on left side, the segment makes 14° with the vertical left side. So the slope of A to B: since left side is vertical, 14° from vertical means the angle with horizontal is 90° - 14° = 76°, so slope is tan(76°).

Similarly, at B, the segment from B to C: C is on top side, and at C, the segment B to C makes 10° with the horizontal top side, so angle with horizontal is 10°, so slope is tan(10°).

At B, the angle φ is the angle between the two segments A to B and B to C.

The direction change can be found from the slopes.

Let the direction of A to B be θ1, with horizontal.

Since at A, the segment makes 14° with vertical, so θ1 = 90° - 14° = 76° from horizontal.

Similarly, the segment B to C makes 10° with horizontal, so θ2 = 10° from horizontal.

Then at B, the angle between the two segments is |θ1 - θ2| = |76° - 10°| = 66°, but that is the smaller angle, but φ is the turn angle, which might be the supplementary or something.

The angle at B for the zig-zag is the angle between the two segments, which is the smaller one, so it could be acute or obtuse.

In the diagram, φ is shown as an acute angle, I think.

But 66° is acute, so φ = 66°? But we have bottom angles, so it's not that simple, and also, we have the bottom part to consider.

Moreover, for the bottom, we have 33° and 26°, so the zig-zag continues.

From C, the segment from B to C is to C, then from C, it goes to the bottom.

At C on top, the segment from B to C arrives, and then the next segment from C to another point.

But at C, for the top side, the angle is 10° for the segment B to C, but the next segment from C will have a different direction.

The angle at C for the top side is given as 10°, which is for the segment B to C, but for the next segment, it could be different.

In the diagram, at C, the angle 10° is labeled, which is the angle between the top side and the segment to B, so for the segment from B to C.

Then from C, the next segment is to a point on the bottom, say D.

At D, the angle is 33° for the bottom side.

Similarly, there is 26° at E.

But to have E, there must be another segment.

Perhaps from C, it goes to D on bottom, then from D to E on bottom? No.

I think I need to include the internal point for the bottom turn.

But there is only one φ.

Perhaps the φ is for the turn at B, and for the bottom, the turn is at the boundary.

Let's assume that from C on top, the segment goes to D on bottom, with angle at D 33°.

Then at D, the segment from C to D makes 33° with the horizontal bottom side.

Since it's arriving at D from above, the direction with horizontal is 33° down from horizontal or something.

Then from D, the next segment goes to a point on the right side, say F, with an angle at F, but not given.

Then from F to E on bottom with angle 26° at E? E on bottom, so from F to E is from right to bottom, so at E, it arrives, and the angle is 26° with the bottom.

But for the bottom side, at D, the angle is 33° for the segment to F, and at E, 26° for the segment from F, but the segment from D to F is not on the bottom; it's diagonal to the right.

Then on the bottom side, between D and E, there is the horizontal segment, but the zig-zag doesn't use it, so D and E are points, and the segment from D to F and from F to E, but F is on right, so at F, there is a turn.

But no angle given at F.

Moreover, the internal point B is only for the top part.

But we have the 26° at E on bottom, which is for the segment from F to E.

But to find φ, we need to consider the whole path.

But it's complicated with many points.

Perhaps for the rectangle, the sum of the angles on the bottom must be 90 degrees.

On the bottom side, there are two points: D with 33° and E with 26°.

At the bottom-left corner, the angle is 90 degrees between bottom and left side.

At the bottom-right corner, 90 degrees between bottom and right side.

At point D, the angle between the bottom side and the zig-zag segment is 33°. This segment is from C to D, so it is arriving at D from above, so the angle with horizontal is 33°, but since it's coming down, the direction is 33° below horizontal.

Similarly, at E, the angle between the bottom side and the zig-zag segment is 26°. This segment is from F to E, so it is arriving at E from above or from the side.

If F is on the right side, then from F to E is diagonal down to the bottom.

The angle at E is 26° with the horizontal, so the segment makes 26° with horizontal.

For the bottom side, from the left corner to D: the side is straight, so no angle.

At D, the angle is 33° for the segment.

Then from D to E: the side is straight horizontal, so no angle for the zig-zag in between.

At E, the angle is 26° for the segment.

Then from E to the right corner: no angle.

So the only angles on the bottom side are 33° at D and 26° at E, but these are the angles with the segments, not with the side itself for turn.

The total contribution for the side to be horizontal is that the angles must be such that the deviations are consistent, but for the sum, it's the angles that the segments make with the side.

But for the rectangle to be closed, the net effect must be zero.

I recall that in such problems, the sum of the angles on one side of the rectangle must equal 90 degrees for the zig-zag to close properly.

For example, consider the top side: the angles on the top side.

On the top side, there is one point C with angle 10°.

At the left corner, the angle is 90 degrees, but it's not an angle from the zig-zag.

The angle at C is 10°, which is the angle between the top side and the segment, so for the top side, this 10° is the only deviation point.

Similarly, at the right end, there might be a corner, so the total for the top side is 10° from the zig-zag, but it should be 90 degrees? I'm confused.

Perhaps it's the sum of the angles at the turns.

Let's think about the change in direction.

Start from the left side at A: the first segment A to B has a direction that makes 14° with the vertical, so with horizontal 76°.

Then at B, the segment B to C makes 10° with horizontal, so the direction changes from 76° to 10°, so the turn angle is 76° - 10° = 66°, which is the external angle or internal.

The angle at B for the zig-zag is the smaller angle between the two segments, which is min|76-10|, 66°, or 180-66=114, but in the diagram, it looks acute, so 66°.

But we have bottom angles, so it's not the full story.

From C, the segment B to C is at 10° with horizontal, but then the next segment from C to D.

At C, for the top side, the angle 10° is for B to C, but the next segment from C to D will have a different angle with the horizontal.

At C, the segment from B to C arrives at C at an angle of 10° with the horizontal, so it is coming in at 10° from horizontal.

Then the next segment from C to D leaves at some angle.

The angle at the vertex C for the zig-zag is the angle between the incoming and outgoing segments.

But at the boundary, the angle with the side is given only for the segment at that point.

For the top side at C, the angle is given as 10° for the segment B to C, which is the arrival segment.

The departure segment from C to D may have a different angle with the top side.

In the diagram, only one angle is given at C, 10°, so it must be that the next segment is not on the top side; it leaves at a different angle.

Similarly for other points.

At C, the segment B to C makes 10° with the top side, so the direction of B to C is 10° from horizontal.

Then the next segment C to D makes an angle with the horizontal, but at C, there is no given angle for C to D with the top side; the 10° is for B to C.

So for the vertex at C, the angle between B to C and C to D is not given.

But in the zig-zag, C is a vertex on the boundary, so the turn angle at C is not specified.

Only for the internal point B, the turn angle is given as φ.

For the bottom points, similar issue.

To simplify, perhaps for the purpose of this problem, the only internal vertex is B, and the other points are on the boundary with given angles for the segments.

But for the bottom, with two points, it's messy.

Let's assume that from C on top, the segment goes directly to a point on the bottom, and the 33° and 26° are for the same segment or something, but that can't be.

Another idea: perhaps the 33° and 26° are the angles at the bottom for the two segments forming the zig-zag at that point.

But at a point on the bottom, there is only one segment leaving, so the angle is with the side.

I think I need to look for the standard method for such problems.

I recall that for a zig-zag line in a rectangle with multiple segments, the sum of the acute angles on the top must equal the sum on the bottom or something.

Let's list all the boundary angles:

• At C on top: 10° ( between top and segment B to C)

• At A on left: 14° ( between left and segment A to B)

• At D on bottom: 33° ( between bottom and segment C to D)

• At E on bottom: 26° ( between bottom and segment E to F or from F to E)

But E is on bottom, so segment to or from E.

Assume that at E, the segment is from above, so angle 26° with horizontal.

But for the bottom side, to have two points, it means that the zig-zag has a "dent" or something.

Perhaps the internal point B is for the top, and for the bottom, there is another internal point, but only one φ is shown.

Let's look at the diagram: in the image, the angle φ is between the segment to 14° and from 10°, but also, the 33° and 26° are on the bottom, and the line from the internal point goes down with two different angles, but it's the same segment.

I think I see: in the diagram, from the internal point B, there is one segment to the top with 10° at C, and one segment to the left with 14° at A, but then from B, there is also a segment down to the bottom, but in the diagram, B is only connected to A and C, so it's a dead end, but the zig-zag must go down.

Perhaps B is not the only internal point; the φ is at B, but there is another vertex.

The diagram shows only one angle marked, φ, so probably only one internal point.

Perhaps the 33° and 26° are for the segment from B to the bottom.

But 33° and 26° are different, so it can't be the same segment.

Unless the segment is not straight, but it is.

I think I have to accept that there are two points on the bottom.

Let's assume the zig-zag has vertices: A on left, B internal, C on top, then from C to D on bottom, then from D to E internal or something, but no other internal.

Perhaps E is on the right side.

But the 26° is on the bottom in the diagram.

Let's read the problem: "Valentin draws a zig-zag line inside a rectangle as shown in the diagram. For that he uses the angles 10°,14°,33° and 26°. How big is angle φ?"

And the diagram has 10°,14°,33°,26° and φ.

From the image: at the top-left, 10°; on the left side, 14°; then φ at the first turn; then on the bottom, 33° and 26°, with 33° on the left and 26° on the right.

Also, the zig-zag line goes from the left side at 14°, down to the internal point, then up to the top at 10°, then from there down to the bottom at 33°, then from there to the right at 26°, but 26° is on the bottom, not on the right.

26° is on the bottom, so it must be that from the point on bottom with 33°, it goes to the point on bottom with 26°, but that would be along the bottom, so not.

Perhaps from the point on bottom with 33°, it goes to a point on the right side, and the 26° is for that point on the bottom for a different segment.

I think the only logical way is that there is a point on the bottom with 33°, and another point on the bottom with 26°, and the segment between them is not part of the zig-zag; the zig-zag has segments to each.

But for the zig-zag to connect, from C on top to D on bottom with 33°, then from D to E on bottom with 26°, but D to E is on the bottom, so the segment is horizontal, so the angle at D for the zig-zag is the angle between C to D and D to E. Since D to E is horizontal, and C to D is coming from above at 33° with horizontal, so the direction of C to D is 33° below horizontal, so at D, the incoming segment has direction 33° below horizontal, and the outgoing segment D to E is horizontal, so the turn angle at D is 33°, but it's not labeled.

Then at E, the segment D to E arrives horizontal, then the next segment from E to F on right with angle 26° at E for the bottom side, so the segment from E to F makes 26° with horizontal.

But the 26° is at E for the segment to F, so it's the departure.

But in the diagram, the 26° is labeled on the bottom, so it's the angle with the bottom side.

But for the zig-zag, at E, the turn angle is between D to E and E to F.

D to E is horizontal, E to F makes 26° with horizontal, so the turn angle is 26°.

But it's not φ.

Moreover, we have the internal point B with φ.

So for the whole thing, to find φ, we need to consider the net effect or the closure.

Perhaps for the rectangle, the sum of the angles on the top and bottom must be related.

Let's consider the horizontal and vertical projections.

Suppose the rectangle has width W and height H.

The first segment from A to B: length L1, at angle 76° to horizontal (since 14° from vertical).

Then from B to C: length L2, at angle 10° to horizontal.

Then from C to D: length L3, at angle 33° to horizontal, but since it's from top to bottom, and 33° at D with horizontal, so the direction is 33° below horizontal.

Then from D to E: length L4, but D to E is along the bottom, so horizontal, but for the zig-zag, if D to E is a segment, it should have a direction, but it's on the bottom, so it can be considered as part of the side, but for the path, it has length, but direction horizontal.

Then from E to F: length L5, at angle 26° to horizontal, but to where? If to the right side, then it might be vertical or at an angle.

But at E, the segment from E to F makes 26° with horizontal, so it could be going up or down.

Since it's from the bottom, and to a point on the right side, it might be going up.

But the angle at E is 26° with the bottom horizontal, so if it's going up, the angle is 26° above horizontal.

Then from E to F at 26° above horizontal to F on right side.

Then from F to the top-right corner or something.

But it's getting too many unknowns.

Moreover, for the right side, there are no given angles.

To close the rectangle, the net horizontal and vertical displacement must be zero, but the zig-zag starts on left and ends on right or something.

Assume the zig-zag starts at A on left and ends at E on bottom or on right.

But with the points, it's not clear.

Perhaps for the purpose of this problem, the angle φ can be found from the given angles using the fact that the sum of the turns must be 360 degrees or for the rectangle, the net turn is 0.

The total turning angle for the path must be 0 since it's a closed shape or from start to end.

The zig-zag line is not closed; it's a line inside the rectangle from one point to another.

The problem doesn't say it's closed; it's just a zig-zag line inside the rectangle.

So the ends are on the boundary.

So the total turning angle from start to end may not be zero.

For example, from A to B to C, the turn at B is φ, then at C, there is a turn to go down, etc.

But the turns at the boundary points are not specified.

At the boundary points, when the path turns at a point on the boundary, the turn angle is constrained by the boundary.

For example, at C on top, when the path arrives from B to C at 10° to horizontal, then it leaves to D at some angle, say α to horizontal.

Then the turn angle at C is |10 - α|, but it's not known.

Similarly at D on bottom, arrives from C to D at 33° to horizontal (below), then leaves to E or to F at some angle.

At E, if on bottom, arrives from D to E horizontal, then leaves to F at 26° to horizontal, etc.

But too many unknowns.

Perhaps for this specific diagram, the only turn with unknown angle is at B, and the other turns can be determined from the given angles.

Let's assume that at C, the next segment is to D on bottom with angle at D 33°, so the segment C to D makes 33° with horizontal, so the direction is 33° below horizontal.

Since it leaves from C at an angle with the horizontal.

At C, the segment B to C arrives at 10° to horizontal, so coming in at 10° from horizontal (since 10° with horizontal, and from B to C, so if B is below, it is coming up at 10° to horizontal).

Then the segment C to D leaves at 33° to horizontal, so going down at 33° to horizontal.

So at C, the turn angle is the change in direction: from incoming 10° to outgoing -33° ( below horizontal), so the change is -33° - 10° = -43°, so a turn of 43° to the right (clockwise) for the path.

But this turn is at the boundary, not specified.

Similarly, at D, the segment C to D arrives at 33° below horizontal, then if it goes to E on bottom with angle 26°, but E is on bottom, so if it goes to E, D to E horizontal, then at D, the turn from incoming 33° below horizontal to outgoing 0° (horizontal), so turn of +33° ( counterclockwise).

Then at E, from D to E horizontal incoming, then to F at 26° above horizontal, so turn of 26°.

But again, not helpful for φ.

I think I found a better way: in many such problems, the key is that the sum of the angles on the top row must equal the sum on the bottom row for the zig-zag to be consistent.

For example, on the top side, the angles given are 10° at C, but there is also the corner.

Perhaps the 10° and 14° are on the top-left, and 33° and 26° on the bottom, and φ is the turn.

Another idea: perhaps the 10° and 14° are the angles at the first vertex, and 33° and 26° at the last, but with the internal turn.

Let's assume that the zig-zag has three segments: from left to internal, from internal to top, then from top to bottom with the given angles.

But then for the bottom, there is only one point with 33°, but 26° is also given, so not.

Perhaps the 26° is for the right side.

In the diagram, 26° is on the bottom, so it must be on the bottom.

I think I need to look for the answer online or think differently.

Perhaps the angle φ can be found by considering the triangle formed.

In the diagram, there is a triangle formed by the zig-zag and the sides.

For example, from the left side, with 14°, and the internal point, and the top with 10°, but also the bottom with 33° and 26°.

Notice that 14° + 10° = 24°, and 33° + 26° = 59°, not equal.

26+14=40, 33+10=43, not equal.

10+26=36, 14+33=47, not.

14+33=47, 10+26=36, not.

Perhaps for the internal angle.

Another thought: at the internal point B, the angle φ is related to the adjacent angles.

For example, the 14° at A and 10° at C, but A and C are not adjacent to B in terms of the rectangle grid.

I recall that in a rectangle with a zig-zag, the sum of the angles on the top must equal the sum on the bottom for the opposite side.

Let's consider the vertical sides.

On the left side, at A, angle 14°, so the segment A to B makes 14° with the vertical, so it is 76° with horizontal.

Then the internal point B, then to C on top at 10° with horizontal.

Then from C to D on bottom at 33° with horizontal.

Then for the bottom, at D, 33° with horizontal, so the segment C to D is 33° with horizontal.

Then from D to E on bottom at 26°, but E is on bottom, so the segment from D to E is not defined.

Assume that from C to D is the only segment to the bottom, and the 26° is for a different point.

Perhaps the 26° is the angle for the segment from D to the right or something.

I think I have to assume that the 26° is at a point on the bottom for the segment to the right side.

But let's assume that at the bottom point with 33°, the segment is to a point on the right side, and the 26° is not used for the zig-zag path, but it is given, so it must be part of it.

Perhaps the 26° is the angle at the bottom for the final segment.

But let's assume the zig-zag ends at E on bottom with the angle 26° for the segment to the right corner or something.

But the right corner has 90 degrees.

To simplify, perhaps for the calculation of φ, the bottom angles are not needed because the turn at B is determined only by the top angles.

But that can't be, because 14° and 10° give 66°, but with the bottom, it might be different.

Let's calculate the turn at B.

From A to B: direction 76° from horizontal.

From B to C: direction 10° from horizontal.

So at B, the path turns from 76° to 10°, a change of 10 - 76 = -66°, so a turn of 66° to the right.

The angle φ at B is the external angle or internal? The angle between the two segments is 66°, which is the smaller angle, so φ = 66°.

But then why are the bottom angles given? Perhaps for verification or because the diagram has them, but for φ, it's 66°.

But 66° is not the typical value, and with the bottom angles, it might be that the turn is not that simple.

Perhaps the 14° and 10° are not with the sides in that way.

Another possibility: the 10° at the top is for the segment from the internal to the top, but the 14° on the left is for the segment from the left to the internal, so the directions are as I had.

Perhaps the internal point B is such that the segment to the bottom is also from B, but in the diagram, it's not.

I think I should go with 66°.

But let's check the numbers: 14° and 10°, 90-14=76, 76-10=66.

Then for the bottom, 33° and 26°, 33+26=59, which is not related to 66.

26+10=36, 33+14=47, not.

14+26=40, 10+33=43, not.

10+14=24, 33+26=59, not equal.

Perhaps the difference: 59-24=35, not related to 66.

Maybe for the rectangle, the sum of the angles on the top must equal the sum on the bottom.

On the top, we have 10° and also the 14° is on the left, but 14° is on the left side, not on top.

The 10° is on top, 14° on left, so for the top side, only 10°, for bottom, 33° and 26°, so 10 vs 59, not equal.

Perhaps the angles are grouped differently.

I think I found a solution online or in my memory: for such a zig-zag, the angle φ can be found as 180 - 10 - 14 - 33 - 26 or something, but 180-10-14-33-26=180-83=97, not.

Or 10+14+33+26=93, 180-93=87, not.

Perhaps for the internal angle, it is 50 degrees or something.

Another idea: perhaps the 33° and 26° are the angles at the bottom for the same point but different segments, but that doesn't make sense.

Let's assume that at the internal point, the segment to the bottom makes an angle with the horizontal, but it's not given.

I think I need to look for the answer.

Perhaps the angle φ is 50 degrees.

Let's assume that the sum of the given angles and φ must be 180 or 90.

10+14+33+26=83, 180-83=97, not φ.

83 + φ = 180, φ=97, but 97 is obtuse, in the diagram it looks acute.

10+14+33+26+φ=180, then 83+φ=180, φ=97.

But 97 is possible.

But let's see the diagram; it looks acute.

Perhaps for the triangle formed.

In the diagram, there is a triangle on the left: with the 14° and the internal, but also the 10° is on top.

From the left side, with 14°, and the top with 10°, and the internal point, the angle at B might be 180 - 14 - 10 = 156 degrees, but that is the external angle or something.

At point B, the angle between the two segments is φ, but in the triangle A-B-C, with A on left, C on top, B internal.

In this triangle, at A, the angle is 14° between the left side and the segment, but the segment A to B is the side, and the left side is one side, but at A, for the triangle, the angle is between the segment A to B and the left side, which is 14°, but the left side is not a side of the triangle; the triangle has sides A to B, B to C, and A to C, but A to C is not a side; it's the boundary.

The points A, B, C are not forming a triangle with A to C; A and C are on different sides.

So the triangle is A-B-C, with sides AB, BC, and CA, but CA is not a straight line; it is the path along the boundary.

So the polygon is A-B-C and then to the boundary.

For the path from A to B to C, the turn at B is φ.

The angle at A between the left side and AB is 14°, but the left side is perpendicular to the top side.

At A, the left side is vertical, AB is at 14° to vertical, so the angle between AB and the horizontal is 76°, as before.

Similarly at C, BC is at 10° to horizontal.

Then for the line from A to C directly, the angle would be different, but not relevant.

I think I have to accept that φ = 66° or 97°, but let's search for a standard approach.

Upon second thought, in the diagram, the 10° and 14° are on the same "corner", and 33° and 26° on the bottom "corner", and the internal turn is between.

Perhaps the angle φ is 50 degrees, as 90 - (10+14+6) or something.

10+14=24, 33+26=59, 59-24=35, 90-35=55, not 50.

50 is 90-40, but 40 not related.

Perhaps the sum of the top angles equals the sum of the bottom angles for the opposite, but 10+14=24 for top-left, 33+26=59 for bottom, not equal.

For the left side, 14°, for the bottom, 33° and 26°, but 14 vs 59.

I give up.

Let's assume that the turn at B is the difference of the angles.

From the left, the segment is at 76° to horizontal, to the top segment at 10° to horizontal, so turn of 66°.

Then for the bottom, the segment from C to D is at 33° to horizontal, and if the last segment is at 26° to horizontal, but it's not connected directly.

Perhaps the path from B to C to D has a turn at C of 10° to 33° = 23° turn, but not used.

I think for the angle φ, it is 66 degrees.

So I'll go with that.
But let's box it.

φ = 90 - 14 + 10? 90-14=76, 76-10=66.

So 66 degrees.

But the problem has 33 and 26, so perhaps it's not.

Perhaps the 14° and 10° are the angles, and 33 and 26 are for the bottom to make the net turn zero.

For example, from the first segment at 76°, to the last segment at some angle.

But the last segment is at 26° to horizontal, for example.

Then the total turn from first to last direction.

From first direction 76°, to last direction, say α.

The total turn is the sum of the turns at the vertices.

At B, turn of 66° (say, to the right).

Then at C, from 10° to 33° below, so from 10° to -33°, change of -43°, turn of 43° to the right.

Then at D, if it goes to E, from -33° to 0°, turn of 33° to the left, +33°.

Then at E, from 0° to 26° above, turn of 26° to the left.

But the last direction is 26° above horizontal.

The first direction is 76°, last is 26°, so the net change in direction is 26 - 76 = -50°, so the total turn should be -50°.

Now, the turns: at B: -66° ( right turn, negative), at C: -43° ( right turn), at D: +33° ( left turn, positive), at E: +26° ( left turn).

Sum of turns: -66 -43 +33 +26 = -109 +59 = -50°, yes, matches.

But this is for the turns at the vertices, but for φ, it is at B, which is -66°, but the angle φ is the magnitude, 66°.

In the sum, it works with the given angles.

So φ = 66°.

The 33° and 26° are used for the turns at C and E, but in the diagram, at C and E, the turns are not φ, so for φ, it is 66°.

So I think it's correct.

So angle φ is 66 degrees.