Congruence and Similarity
Congruent and similar triangles
Two triangles are congruent when they are the same shape and the same size: one can be picked up and placed exactly onto the other by a rotation, a reflection or a translation, with nothing stretched or shrunk. Two triangles are similar when they are the same shape but possibly a different size: one is an enlargement of the other by a scale factor. In both cases the corresponding angles stay equal, so the triangles look alike; the difference is that congruence also fixes the lengths, while similarity allows every side to grow or shrink by the same factor.
The congruence tests
You do not need to measure all six parts of a triangle to be sure two triangles match. Four minimal sets of facts each pin a triangle down completely. SSS uses the three sides. SAS uses two sides and the angle between them, the included angle. ASA uses two angles and a side. RHS uses a right angle, the hypotenuse and one other side. To show two triangles are congruent you look at the information you are given and name the single test that it matches; once one test is satisfied the two triangles must be congruent.
Why SSA is not a test
It is tempting to think that two sides and any angle are enough, but two sides and a non-included angle, written SSA, do not work. With the angle sitting away from the gap between the two sides, the third side can often swing to two different positions, producing two different triangles from the very same measurements. This is called the ambiguous case, and it is the reason SSA is not on the list of tests. It also explains why the angle in SAS must be the included one, the angle held between the two named sides, so that the triangle is locked in place.
The similarity tests
Triangles are similar under three matching conditions. If the corresponding angles are equal the triangles are similar; this is the AAA test, and because the three angles of any triangle always add to the same total, knowing two equal angles is already enough. If the corresponding sides are in the same ratio, such as 3:6 = 4:8 = 5:10, the triangles are similar by the SSS similarity test. If two pairs of sides are in the same ratio and the included angles are equal, that is the SAS similarity test. Each condition guarantees the same shape while allowing a change of size.
Using a scale factor
In similar triangles every pair of corresponding sides shares one number, the scale factor. To move from the small triangle to the large one you multiply each side by the scale factor; to move back you divide. Suppose two triangles are similar with a scale factor of 2. A side of 3 in the small triangle matches a side of 6 in the large one, since 3 × 2 = 6. If another side of the small triangle is unknown and its match in the large triangle is 8, divide to undo the factor: 8 ÷ 2 = 4, so the missing side is 4.
Beyond triangles
The same two ideas reach every shape. Two shapes are congruent when translations, rotations and reflections can move one exactly onto the other, and similar when an enlargement, with or without those moves, maps one onto the other. For shapes other than triangles, equal sides alone are not enough: a square and a rhombus with the same side length have equal sides but different angles, so they are not congruent. Two polygons are similar only when every pair of corresponding angles is equal and every pair of corresponding sides shares one scale factor. Some families are always similar, such as any two circles, any two squares or any two equilateral triangles. Rectangles are not: a 2 by 4 rectangle and a 3 by 4 rectangle have equal angles, but their sides do not share one scale factor.
Why this matters
Congruence and similarity sit underneath a great deal of everyday mathematics: repeating pattern and design, scale drawings, maps, models, and the indirect measuring of heights and distances when similar triangles let a short shadow stand in for a tall tree. Knowing the tests means you can decide, from the smallest amount of information, whether two shapes truly match. The common slips are reaching for SSA, which is not a test, and pairing up sides that do not correspond. This unit stays with the named tests and the scale factor; formal proof and trigonometry build on these ideas in later years.