Exercise 128 Page 627

If the triangles are congruent they have the same shape and size. In order for the triangles to have the same shape, they must have at least two pairs of congruent angles. Examining the diagram, we see that this is the case which means we can claim similarity by the Angle-Angle (AA) Similarity Theorem.

If they also have the same size — making them congruent — they must have at least one pair of corresponding sides that are congruent. Again, examining the diagram, we see that $\overline{AB}\cong \overline{TP}.$ Since they are between the same two pairs of congruent angles, the sides are corresponding. Therefore, we can claim congruence by the Angle-Side-Angle (ASA) Congruence Theorem

Rigid Transformations

Examining the diagram, we see that the triangles have different positions and orientations. Let's first translate $△ABC$ so that two corresponding vertices map onto each other.

Next, we rotate $△A^{\prime }{B}^{\prime }{C}^{\prime }$ clockwise until the triangles map onto each other.