Question

1. for each pair of shapes, decide whether or not the two shapes are congruent. explain your reasoning.

Explanation:

Step1: Check side - lengths

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or count grid - squares for horizontal and vertical sides. For polygon \(ABCD\), \(AB=\sqrt{(2 - 1)^2+(3 - 2)^2}=\sqrt{1 + 1}=\sqrt{2}\), \(BC = 2\), \(CD=\sqrt{(3 - 1)^2+(1 - 3)^2}=\sqrt{4 + 4}=2\sqrt{2}\), \(DA=\sqrt{(1 - 3)^2+(2 - 1)^2}=\sqrt{4+1}=\sqrt{5}\). For polygon \(EFGH\), \(EF=\sqrt{(2 - 1)^2+(1 - 0)^2}=\sqrt{1 + 1}=\sqrt{2}\), \(FG = 2\), \(GH=\sqrt{(3 - 1)^2+(3 - 1)^2}=\sqrt{4 + 4}=2\sqrt{2}\), \(HE=\sqrt{(1 - 3)^2+(0 - 1)^2}=\sqrt{4 + 1}=\sqrt{5}\).

Step2: Check angles

We can observe that the angles of the two polygons match. Since we can translate and rotate one polygon to exactly overlap the other, and all corresponding side - lengths are equal and all corresponding angles are equal.

Answer:

The two shapes are congruent.