Question

which statements are true about polygons? select three options. all sides and all angles in a polygon are congruent. the sides of a polygon are segments that intersect exactly two other segments, one at each endpoint. in a polygon, all segments with a common endpoint are collinear. if all of the sides of a convex polygon are extended, none of them will contain any points that are inside the polygon. the extension of at least one side or diagonal in a concave polygon will contain a point that is inside the polygon.

Explanation:

Step1: Analyze the first option

A polygon is not necessarily regular (all - sides and all - angles congruent). For example, a rectangle is a polygon with all angles equal but adjacent sides are not equal. So, "All sides and all angles in a polygon are congruent" is false.

Step2: Analyze the second option

By the definition of a polygon, each side (edge) is a line segment. Each vertex (endpoint of a side) is shared by exactly two sides. So, "The sides of a polygon are segments that intersect exactly two other segments, one at each endpoint" is true.

Step3: Analyze the third option

If all segments with a common endpoint were collinear, the figure would be a line segment, not a polygon. A polygon has angles at its vertices, which means the adjacent sides at a vertex are not collinear. So, "In a polygon, all segments with a common endpoint are collinear" is false.

Step4: Analyze the fourth option

For a convex polygon, by definition, if we extend any of its sides, the extended line will not pass through the interior of the polygon. So, "If all of the sides of a convex polygon are extended, none of them will contain any points that are inside the polygon" is true.

Step5: Analyze the fifth option

For a concave polygon, there is at least one interior angle greater than \(180^{\circ}\). When we extend the side(s) adjacent to this angle or a diagonal, the extension will pass through the interior of the polygon. So, "The extension of at least one side or diagonal in a concave polygon will contain a point that is inside the polygon" is true.

Answer:

B. The sides of a polygon are segments that intersect exactly two other segments, one at each endpoint; D. If all of the sides of a convex polygon are extended, none of them will contain any points that are inside the polygon; E. The extension of at least one side or diagonal in a concave polygon will contain a point that is inside the polygon.