Question

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

Explanation:

Step1: Recall congruence definition

Two shapes are congruent if they have the same size and shape, which means corresponding sides and angles are equal.

Step2: Analyze side - lengths

Count the grid - squares for side - lengths. For polygon \(ABCD\), \(AB=\sqrt{( - 2+3)^2+(4 - 3)^2}=\sqrt{1 + 1}=\sqrt{2}\), \(BC = 1\), \(CD=\sqrt{( - 3+4)^2+(2 - 1)^2}=\sqrt{1 + 1}=\sqrt{2}\), \(DA=\sqrt{( - 4+3)^2+(1 - 2)^2}=\sqrt{1 + 1}=\sqrt{2}\). For polygon \(FGHIJ\), \(FG=\sqrt{(1 - 0)^2+( - 1+1)^2}=1\), \(GH = 1\), \(HI=\sqrt{(2 - 1)^2+(1 - 1)^2}=1\), \(IJ=\sqrt{(2 - 1)^2+(1 - 0)^2}=\sqrt{1 + 1}=\sqrt{2}\), \(JF=\sqrt{(0 - 1)^2+( - 1-0)^2}=\sqrt{1 + 1}=\sqrt{2}\). The side - lengths do not match exactly for all corresponding sides.

Step3: Analyze angles

We can also use the slope of the sides to check angles. The slope of \(AB\) is \(m_{AB}=\frac{4 - 3}{-2+3}=1\), the slope of \(FG\) is \(m_{FG}=0\). The angles between the sides are different.

Answer:

The two shapes are not congruent.