Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, the AA postulate states that two

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s are similar if they have two corresponding

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

. The AA postulate follows from the fact that the sum of the

interior angle In geometry, an angle of a polygon is formed by two adjacent sides. For a simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point withi ...

s of a

is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180-(32+64)=84. (This is sometimes referred to as the AAA Postulate—which is true in all respects, but two angles are entirely sufficient.) The postulate can be better understood by working in reverse order. The two triangles on grids A and B are similar, by a 1.5

dilation wiktionary:dilation, Dilation (or dilatation) may refer to: Physiology or medicine * Cervical dilation, the widening of the cervix in childbirth, miscarriage etc. * Coronary dilation, or coronary reflex * Dilation and curettage, the opening of ...

from A to B. If they are aligned, as in grid C, it is apparent that the angle on the origin is congruent with the other (D). We also know that the pair of sides opposite the origin are parallel. We know this because the pairs of sides around them are similar, stem from the same point, and line up with each other. We can then look at the sides around the parallels as transversals, and therefore the corresponding angles are congruent. Using this reasoning we can tell that similar triangles have congruent angles. Now, because this article is practically over, you might want to know what AA postulate can be used for. It is used proving the

Angle Bisector Theorem In geometry, the angle bisector theorem is concerned with the relative lengths of the two Line segment, segments that a triangle's side is divided into by a Line (geometry), line that Bisection, bisects the opposite angle. It equates their relat ...

. AA postulate is one of the many similarity ways for determining similarity in a triangle.

References

* http://hanlonmath.com/pdfFiles/464Chapter7Sim.Poly.pdf ''(Unused Source)'' {{DEFAULTSORT:Aa Postulate Elementary geometry Triangle geometry Euclidean plane geometry