Geometry Right Triangles and Similarity: Theorems, Postulates and Corollaries – Flashcards
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If two angles of ones triangle are congruent to two angles of another triangle, then the triangles are similar.
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AA Similarity Postulate
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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.
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SAS Similarity Theorem
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If the sides of two triangles are in proportion, then the triangles are similar.
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SSS Similarity Theorem
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If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
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Triangle Proportionality Theorem
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If three parallel lines intersect two transversals, then they divide the transverals proportionally
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Proportionality Corollary
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If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.
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Triangle Angle-Bisector Theorem
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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.
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Right Triangle Similarity Theorem 1
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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.
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Right Triangle Similarity Corollary 1
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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.
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Right Triangle Similarity Corollary 2
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In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
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Pythagorean Theorem
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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.
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Converse of the Pythagorean Theorem
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If c^2 = a^2 + b^2, then the measure of angle C = 90 and the triangle is right.
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Right or Not? Right
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If c^2 < a^2 + b^2, then the measure of angle C < 90 and the triangle is acute.
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Right or Not? Acute
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If c^2 > a^2 + b^2, then the measure of angle C > 90 and the triangle is obtuse.
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Right or Not? Obtuse
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In a 45-45-90 triangle, the hypotenuse is the root 2 times as long as a leg.
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45-45-90 Theorem
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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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30-60-90 Theorem
