In context of the adjoining figure, answer the following questions:

(i) Is ABCDEFG a polygon?

(ii) How many sides does it have?

(iii) How many vertices does it have?

(iv) Are $\overline{AB} \text{ and } \overline{FE}$ adjacent sides?

(v) Is $\overline{GF}$ a diagonal of the polygon?

(vi) Are $\overline{AC}, \overline{AD} \text{ and } \overline{AE}$ diagonals of the polygon?

(vii) Is point P in the interior of the polygon?

(viii) Is point A in the exterior of the polygon?

(i) The figure ABCDEFG is a simple closed curve made up entirely of line segments. Therefore, it satisfies the definition of a polygon.

∴ Yes, ABCDEFG is a polygon.

(ii) The sides of the polygon ABCDEFG are $\overline{AB}, \overline{BC}, \overline{CD}, \overline{DE}, \overline{EF}, \overline{FG} \text{ and } \overline{GA}$. So, the polygon has 7 sides.

∴ The polygon ABCDEFG has 7 sides.

(iii) The vertices of the polygon ABCDEFG are A, B, C, D, E, F and G. So, the polygon has 7 vertices.

∴ The polygon ABCDEFG has 7 vertices.

(iv) Two sides of a polygon with a common end point are called adjacent sides.

The side $\overline{AB}$ has end points A and B, and the side $\overline{FE}$ (same as $\overline{EF}$) has end points F and E. They do not share any common end point, so they are not adjacent sides.

∴ No, $\overline{AB} \text{ and } \overline{FE}$ are not adjacent sides.

(v) The diagonals of a polygon are the line segments formed by joining non-adjacent vertices.

The segment $\overline{GF}$ (same as $\overline{FG}$) joins the vertices G and F, which are adjacent vertices (they are the end points of the same side $\overline{FG}$). So, $\overline{GF}$ is a side of the polygon, not a diagonal.

∴ No, $\overline{GF}$ is not a diagonal of the polygon; it is a side of the polygon.

(vi) Let us check each line segment:

• In $\overline{AC}$: vertex A is adjacent to B and G, but not to C. So, A and C are non-adjacent vertices.
• In $\overline{AD}$: vertex A is adjacent to B and G, but not to D. So, A and D are non-adjacent vertices.
• In $\overline{AE}$: vertex A is adjacent to B and G, but not to E. So, A and E are non-adjacent vertices.

Since each of these segments joins two non-adjacent vertices, they are all diagonals of the polygon.

∴ Yes, $\overline{AC}, \overline{AD} \text{ and } \overline{AE}$ are all diagonals of the polygon.

(vii) Looking at the figure, point P does not lie inside the boundary of polygon ABCDEFG.

∴ No, point P is not in the interior of the polygon.

(viii) Point A is a vertex of the polygon ABCDEFG, so it lies on the boundary of the polygon, not in its exterior.

∴ No, point A is not in the exterior of the polygon; it lies on its boundary.