In geometry, congruence allows you to swap the locations of figures without altering other properties such as distances and angles. For example, a triangle could be rotated and translated to produce another triangle identical to the first, and the two could even be reflected on a line to match up their corresponding vertices.
To prove that two figures are congruent, you need to establish equality of at least three corresponding parts of the shapes. In most cases, this is enough. But you might also use the SSS, ASA or RHS (right-angle-hypotenuse-side) congruence criteria.
SAS (side-angle-side): If two pairs of adjacent sides of two triangles are equal in length, then the triangles are congruent. This is one of the postulates in most systems of axioms.
ASA (angle-side-angle): If two pairs of included angles of two triangles are equal in measure, then the triangles are congruent.
RHS (right-angle-hypotenuse-side): If the hypotenuses of two right-angled triangles are equal in length, then the angles of the triangles are congruent. This criterion is the basis for many congruence theorems in elementary geometry.
So which of these triangle pairs can be mapped to each other using a single translation? The pair ABDC and BCD can be mapped to each other by a translation that involves moving one triangle up to the left. This works because the triangles have the same vertices and the side lengths can be verified by the distance formula. In addition, the pair DABC and DPQR can be mapped to each other by reversing the direction of one of the translations. This is because the triangles have the same vertices, and the side lengths can be verified by matching up the triangles with the same vertices.