Question

1. this design began from the construction of a regular hexagon.

a. draw 1 segment so the diagram has another hexagon that is congruent to hexagon abcihg.
b. explain why the hexagons are congruent.
(from unit 1, lesson 22.)

Explanation:

a.

Step1: Analyze the structure

The original hexagon \(ABCIHG\) is part of a larger regular - hexagon - based design. To create a congruent hexagon, we can use the symmetry of the regular hexagon.

Step2: Draw the segment

Draw a segment from \(K\) to \(H\). The new hexagon \(HKJILG\) (formed by the vertices \(H\), \(K\), \(J\), \(I\), \(L\), \(G\)) is congruent to hexagon \(ABCIHG\).

b.

1. Side - length equality: In a regular hexagon, all side lengths are equal. The segments that form the sides of the new hexagon (e.g., \(HK\), \(KJ\), \(JI\), \(IL\), \(LG\), \(GH\)) are equal in length to the sides of hexagon \(ABCIHG\) (e.g., \(AB\), \(BC\), \(CI\), \(IH\), \(HG\), \(GA\)) because of the properties of the original regular - hexagon construction.
2. Angle equality: All interior angles of a regular hexagon are equal (\(120^{\circ}\)). Since the new hexagon is formed within the same regular - hexagon - based structure, the angles of the new hexagon are equal to the angles of hexagon \(ABCIHG\).
3. Congruence criteria: By the Side - Angle - Side (SAS), Side - Side - Side (SSS) congruence criteria for polygons (in the case of a hexagon, if all corresponding sides and angles are equal), the two hexagons are congruent.

Answer:

a. Draw a segment from \(K\) to \(H\) (the new hexagon is \(HKJILG\)).
b. The hexagons are congruent because all corresponding side lengths are equal (due to the regular - hexagon construction) and all corresponding interior angles are equal (interior angles of a regular hexagon are \(120^{\circ}\)), satisfying the congruence criteria for polygons.