What is meant when two plane figures are described as being "congruent"?

Name the four cases under which two triangles can be proven to be congruent.

In △ABC and △PQR, if AB=PQ, BC=QR, and AB^C=PQ^​R, under which case are the two triangles congruent?

In △ABC and △PQR, if AB^C=PQ^​R, AC^B=PR^Q, and AC=PR, under which case are the two triangles congruent?

In △ABC and △PQR, if AB=PQ, BC=QR, and AC=PR, under which case are the two triangles congruent?

In the right-angled triangles △ABC and △PQR, where B^ and Q^​ are right angles, if hypotenuse AC=PR and side AB=PQ, under which case are the two triangles congruent?

In quadrilateral ABCD, AB=AD and the diagonal AC bisects the angle BA^D. Prove that △ABC≡△ADC.

In △ABC, the angle bisector of BA^C meets BC at X. If AB^X=AC^X, prove that △ABX≡△ACX.

The opposite sides of the quadrilateral PQRS are equal in length (PQ=RS, PS=QR). Prove that △PSR≡△RQP.

A circle has its centre at O. A perpendicular line from O meets a chord AB at point X. Prove that △OXA≡△OXB.

If △ABC≡△PQR, what can be concluded about the lengths of the corresponding sides BC and QR, and the magnitudes of the corresponding angles BA^C and QP^R?

In quadrilateral ABCD, diagonals AC and BD bisect each other at O. Prove that the opposite sides AD and BC are parallel to each other.

In △PQR, perpendiculars QY and RX are drawn from Q and R to the opposite sides such that QY=RX. Prove that △XQR≡△YRQ.

The equilateral triangles BCF and DCE are drawn on the sides BC and DC of a square ABCD, lying outside the square. Prove that △EDA≡△ABF.

In a parallelogram ABCD, the mid-point of side DC is P. If AD and BP produced meet at E, prove that AB = BQ, where Q is the intersection of DE produced and AB produced.