For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.

1.  

   

   Two figures E F G H and A B C D on a coordinate plane. Figure H G F E is the image of figure A B C D after rotation of 180 degrees, followed by a translation left 3 and down 2 units. A B C D has the coordinates A(5 comma 0), B(4 comma 2), C(3 comma 1) and D(1 comma 1). H G F E has the coordinates H(negative 6 comma 0), G(negative 5 comma negative 2), F(negative 4 comma negative 1 and E(negative 2 comma negative 1).

2.  

   

   Two pentagons on a coordinate plane. Pentagon ABCDE is non-convex and has coordinates A(negative 4 comma 0), B(negative 2 comma 1), C(negative two comma 4), D(negative 3 comma 3) and E(negative 4 comma 3). Pentagon FGHIJ is convex and has coordinates F(1 comma 2), G(3 comma 1), H(4 comma 2), I(4 comma 4) and J(2 comma 4).

3.  

   

   Two triangles D E F and A B C on a coordinate plane. Triangle D E F is the image of triangles A B C after rotation of 90 degrees, followed by a translation left 3 and down 2 units. Triangle A B C has the coordinates A(2 comma negative 2), B(6 comma 0) and C(6 comma 2). Triangle D E F has the coordinates D(negative 1 comma 0), E(negative 3 comma 4) and F(negative 5 comma 4).

4.  

   

Take turns with your partner to decide whether Shape A is congruent to Shape B.

• For each pair of shapes that you decide is congruent or not congruent, explain to your partner how you know.
• For each pair of shapes that your partner decides is congruent or not congruent, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

1.  

   

   Two figures A and B on a coordinate plane, origin L. Figure A has points G(negative 5 comma negative 1), H(negative 5 comma 1), I(negative 2 comma 1), J(negative 1 comma 2), K(0 comma 2) and L(0 comma 0). Figure B has points P(6 comma 5), Q(8 comma 5), R(8 comma 2), S(9 comma 1), T(9 comma 0) and U(7 comma 0).

2.  

   

   Two figures A and B on a coordinate plane. Figure A has points G(negative 4 comma 3), H(negative 1 comma  3), I(negative 1 comma 0), J(negative 3 comma 0), K(negative 3 comma 1) and L(negative 4 comma 1). Figure B has points P(3 comma negative 1), Q(3 comma 2), R(6 comma 2), S(6 comma 1), T(4 comma 1) and U(4 comma negative 1).

3.  

   

   Two figures A and B on a coordinate plane. Figure A has points G(negative 4 comma 3), H(negative 1 comma  3), I(negative 1 comma 0) and J(negative 3 comma 0). Figure B has points P(3 comma negative 1), Q(3 comma 2), R(6 comma 2) and S(6 comma 1).

4.  

   

   Two figure, rhombus A and square B on a coordinate plane. Figure A is a rhombus with side lengths 5 units and has points G(negative 9 comma 1), H(negative 5 comma 4), I(negative 1 comma 1) and J(negative 5 comma negative 2). Figure B is a square with side lengths 5 units and has points P(2 comma 1), Q(7 comma 1), R(7 comma 6) and S(2 comma 6).

5.  

   

   Two figures, square A and rhombus B on a coordinate plane. Figure A is a square with side lengths 3 units and has points G(negative 5 comma 1), H(negative 2 comma 1), I(negative 2 comma 4) and J(negative 5 comma 4). Figure B is a rhombus with side lengths near 2 point 8 units and has points P(3 comma 3), Q(5 comma 5), R(7 comma 3) and S(5 comma 1).

Your teacher will give you a set of four objects.

1. Make a quadrilateral with your four objects and record what you have made.

2. Compare your quadrilateral with your partner’s. Are they congruent? Explain how you know.

3. Repeat Steps 1 and 2, forming different quadrilaterals. If your first quadrilaterals were not congruent, can you build a pair that is? If your first quadrilaterals were congruent, can you build a pair that is not? Explain.