In a triangle, the remote interior angles of an exterior angle are the two non-adjacent interior angles that don't share its vertex. In this case, angles 4 and 6 are the remote interior angles of angle 1.
In the provided image, we see a triangle with an exterior angle labeled ∠1 and several other angles labeled. We are asked to identify the remote interior angles of ∠1.
Recall that:
An exterior angle of a triangle is formed when one side of the triangle is extended.
The remote interior angles of an exterior angle are the two non-adjacent interior angles of the triangle that do not share a vertex with the exterior angle.
Therefore:
Angles 4 and 6 are the remote interior angles of angle 1.
Here's why:
Observe that angle 1 is formed by extending side 1 of the triangle.
Angles 4 and 6 are both interior to the triangle and do not share a vertex with angle 1.
Any other interior angles (2, 3, or 5) share a vertex with angle 1 and thus are not considered remote interior angles.
In summary:
The remote interior angles of ∠1 are angle 4 and angle 6. This is because they are both non-adjacent interior angles of the triangle that do not share a vertex with the exterior angle 1.