Question

You are given two pairs of triangles. For the first pair of triangles, each side and angle of one triangle is congruent to the corresponding side and angle of the other. You show that rigid motions can transform one triangle so that it matches up with the other. For the second pair of triangles, you show that rigid motions can transform one triangle so that each angle or side of one triangle matches exactly with a corresponding angle or side of the other triangle. What have you proved?

Answer 1

Options:

a) If corresponding pairs of sides and corresponding pairs of angles of two triangles are congruent, then the triangles can be matched up exactly using rigid motions.

b) If two triangles can be matched up exactly using rigid motions, then the corresponding pairs of sides and corresponding pairs of angles of the triangles are congruent.

c) Two triangles can be matched up exactly using rigid motions if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

d) If corresponding pairs of sides and corresponding pairs of angles of two triangles are not congruent, then the triangles are not congruent.

Answer:

c) Two triangles can be matched up exactly using rigid motions if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explanation:

For both pairs of triangles, what you proved is how to use rigid motions (i.e. rigid transformations) to make congruent shapes.

When rigid transformation is applied to a shape, the image (i.e. result) of the transformation produces an exact shape (i.e. equal corresponding angles and corresponding sides), meaning that the side lengths and the angles of the preimage (before transformation) and the image (after transformation) is unaltered.

Option (c) is true