School curriculum· 30 min· 20 questions
Learn first: Triangles and their elements
Proving that two triangles are congruent or similar is the main tool of geometry. Congruence lets you set an unknown segment or angle equal to a known one; similarity lets you set up a proportion and calculate the unknown. The properties of an isosceles triangle, the midline, one of the proofs of Pythagoras’ theorem, map scales and measuring a height from a shadow all rest on these two ideas. This lesson covers the tests, how to apply them, Thales’ theorem, the angle bisector theorem and similarity in a right-angled triangle.
Two triangles are congruent if one can be placed exactly on the other (by a translation, rotation or reflection). Congruent triangles have equal corresponding sides and equal corresponding angles. The order of the correspondence matters: △ABC = △KLM means A ↔ K, B ↔ L, C ↔ M.
To prove congruence you need not check all six elements — three are enough:
1) Two sides and the included angle (SAS)
2) One side and the two angles adjacent to it (ASA)
3) Three sides (SSS)
For right-angled triangles there are extra tests: two legs; a leg and an acute angle; hypotenuse and a leg (RHS); hypotenuse and an acute angle.
Note: "two sides and an angle not between them" is not a test in general: two different triangles can fit such data. Three equal angles are not a test either — they give only similarity.
Segments AB and CD meet at O, and each is bisected by that point. Show that AC = BD.
Many well-known properties are proved with the congruence tests:
Order of a proof: choose two triangles containing the required segment or angle → find three equal elements in them (given data, vertically opposite angles, a common side, angles at parallel lines) → name the test → state the equality of the corresponding elements.
A common side is a "free" equal element for two triangles
Triangles whose angles are equal in pairs and whose corresponding sides are proportional are similar: a₁/a = b₁/b = c₁/c = k, where k is the scale factor. Similar figures have the same shape and differ only in size.
Tests: 1) two angles equal; 2) two sides proportional and the included angles equal; 3) three sides proportional
Ratio of perimeters k, of areas k², of volumes k³
Why does area change by k²? Area is a product of two lengths (½ · base · height), each multiplied by k. Corresponding elements (median, height, bisector, inradius and circumradius) are also in the ratio k.
The most used test is two angles: a line parallel to a side of a triangle always cuts off a similar triangle (corresponding angles are equal).
Similar triangles have perimeters 12 and 18. The smaller has area 20. Find the area of the larger.
A line parallel to AC meets AB at M and BC at N; BM = 4, MA = 2, AC = 9. Find MN.
Parallel lines crossing the arms of an angle cut off proportional segments on them
If equal segments are marked on one arm, the segments on the other arm are equal too. This gives the midline property of a triangle: the segment joining the midpoints of two sides is parallel to the third side and half as long (k = 1/2).
Angle bisector theorem: the bisector of an angle of a triangle divides the opposite side in the ratio of the adjacent sides:
BD/DC = AB/AC (AD bisects angle A)
AB = 6, AC = 9, BC = 10. Into what parts does the bisector of angle A divide BC?
A 2 m post casts a 3 m shadow while a tree casts a 15 m shadow. Find the height of the tree.
The height to the hypotenuse divides the triangle into two triangles similar to each other and to the original (each shares an acute angle with it). Hence:
h² = a_c · b_c, a² = c · a_c, b² = c · b_c
Adding the last two gives a² + b² = c(a_c + b_c) = c² — a proof of Pythagoras’ theorem by similarity.
The height divides the hypotenuse into segments 4 and 9. Find the height and the legs.
The triangles formed by the diagonals of a trapezium next to its bases are similar, with k = a/b (the ratio of the bases). If the bases are 4 and 10, the diagonals divide each other in the ratio 2 : 5, and the areas of those triangles are in the ratio 4 : 25.
In practice, similarity is used in maps and drawings (scale), when enlarging photographs and when measuring inaccessible distances and heights. For example, on a map with scale 1 : 50 000, 4 cm corresponds to 2 km on the ground.