Geometry Right Triangles and Similarity: Theorems, Postulates and Corollaries

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AA Similarity Postulate

If two angles of ones triangle are congruent to two angles of another triangle, then the triangles are similar.

SAS Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

SSS Similarity Theorem

If the sides of two triangles are in proportion, then the triangles are similar.

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Proportionality Corollary

If three parallel lines intersect two transversals, then they divide the transverals proportionally

Triangle Angle-Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.

Right Triangle Similarity Theorem 1

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

Right Triangle Similarity Corollary 1

When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

Right Triangle Similarity Corollary 2

When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

Converse of the Pythagorean Theorem

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Right or Not? Right

If c^2 = a^2 + b^2, then the measure of angle C = 90 and the triangle is right.

Right or Not? Acute

If c^2 < a^2 + b^2, then the measure of angle C < 90 and the triangle is acute.

Right or Not? Obtuse

If c^2 > a^2 + b^2, then the measure of angle C > 90 and the triangle is obtuse.

45-45-90 Theorem

In a 45-45-90 triangle, the hypotenuse is the root 2 times as long as a leg.

30-60-90 Theorem

In a 30-60-90 triangle, the hypotenuse is 2x the shortest side and the longest side is root 3 times as the shortest side.

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