Question

8 from unit 2, lesson 1 when rectangle abcd is reflected across line ef, the image is rectangle badc. how do you know that segment ad is congruent to segment bc? a a rectangle has 2 pairs of parallel sides. b any 2 sides of a rectangle are congruent. c corresponding parts of congruent figures are congruent. d congruent parts of congruent figures are corresponding. 9 from unit 1, lesson 22 this design began from the construction of a regular hexagon. describe a rigid motion that will take the figure onto itself.

Explanation:

Question 8

• Option A: Just having parallel sides does not directly prove congruence.
• Option B: Not any two sides of a rectangle are congruent (only opposite sides).
• Option C: Since rectangle \(ABCD\) and its image \(BADC\) are congruent (reflection is a congruence transformation), by the property that corresponding parts of congruent figures are congruent, \(AD\) (a side of \(ABCD\)) and \(BC\) (the corresponding side of \(BADC\)) are congruent.
• Option D: The statement is not in the correct logical order for proving the congruence of \(AD\) and \(BC\) in this context.

A regular hexagon has rotational symmetry of order 6 (rotational symmetry of \(60^{\circ},120^{\circ},180^{\circ},240^{\circ},300^{\circ},360^{\circ}\)). Looking at the figure (which is based on a regular hexagon), a rotation of \(180^{\circ}\) about the center of the hexagon (the mid - point of the line segment joining \(A\) and \(D\) or \(B\) and \(E\) etc.) will map the figure onto itself. Because a \(180^{\circ}\) rotation is a rigid motion (it preserves distances and angles), and for a figure with the symmetry of a regular hexagon (the base of the design), a \(180^{\circ}\) rotation will map each point \(P(x,y)\) to a point \(P'(-x, - y)\) (in a coordinate system centered at the center of the hexagon - like symmetry center) and will map the shaded regions and non - shaded regions onto their congruent counterparts.

Answer:

C. Corresponding parts of congruent figures are congruent.

Question 9