Geometry: Congruence



Proving Similarity of Triangles

Problem : If triangle ABC is similar to triangle DEF, which side is proportional to side BC?

Side EF.

Problem : Triangles ABC and DEC are similar, and line l is parallel to segment AB. What is the length of CE?


Problem : Can two triangles be similar but not congruent? Can they be congruent but not similar?

Every pair of congruent triangles are similar, but every pair of similar triangles isn't necessarily congruent.

Problem : When a line is drawn through an equilateral triangle such that it is parallel to one side and intersects one time with each of the other two sides, how many similar pairs of triangles are created?

Thirty-six. If the equilateral triangle is triangle ABC, and the line intersects with sides AC and BC, then a new triangle, DFC, is created. Let point D be the intersection point of the line and side AC and let point E be the intersection point of side BC. Then triangle DFC is similar to triangles ABC, BCA, CAB, ACB, BAC, and CBA. The five other triangles, FCD, CDF, DCF, FDC, and CFD are also all similar to these six triangles, making 36 pairs of similar triangles.