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When two triangles are exactly equal: the SAS, ASA, SSS and RHS rules, the properties of an isosceles triangle, and the inequalities between the sides and angles of a triangle.
Similar figures and the two conditions for similar triangles. Then Thales' theorem, its converse, and the way a line parallel to one side cuts the other two in the same ratio.
The AA, SSS and SAS criteria for similarity. Then the ratio of the areas of similar triangles, and the Pythagoras theorem and its converse proved by similar triangles.