Theorems & Corollaries (Full) – Flashcards
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Theorem 1
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If A-B-C, then AB+BC=AC.
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Theorem 2
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If OA-OB-OC, then AOB+BOC=AOC.
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Theorem 3
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Complements of the same angle are equal.
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Theorem 4
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Supplements of the same angle are equal.
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Theorem 5
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The angles in a linear pair are supplementary.
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Theorem 6
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Vertical angles are equal.
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Theorem 7
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Perpendicular lines form 4 right angles.
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Theorem 8
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If the angles in a linear pair are equal, their sides are perpendicular.
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Theorem 9
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If two sides of a triangle are equal, the angles opposite them are equal.
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Corollary to Theorem 9
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An equilateral triangle is equiangular.
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Theorem 10
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If two angles of a triangle are equal, the sides opposite them are equal.
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Corollary to Theorem 10
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An equiangular triangle is equilateral.
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Theorem 11 (SSS Theorem)
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If 3 sides of a triangle are equal to 3 sides of another, the triangles are congruent.
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Theorem 12 (Exterior Angle Theorem)
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An exterior angle of a triangle is greater than either remote interior angle.
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Theorem 13
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If two sides of a triangle are unequal, their opposite angles are unequal in the same order.
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Theorem 14
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If two angles of a triangle are unequal, their opposite sides are unequal in the same order.
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Theorem 15
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The sum of any two sides of a triangle is greater than the third.
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Theorem 16
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In a plane, two points equidistant from the endpoints of a line segment determine the segment's perpendicular bisector.
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Theorem 17
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Equal corresponding angles means parallel lines.
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Corollary 1 to Theorem 17
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Equal alternate interior angles means parallel lines.
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Corollary 2 to Theorem 17
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Supplementary interior angles on the same side of the transversal means parallel lines.
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Corollary 3 to Theorem 17
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In a plane, two lines perpendicular to a third line are parallel.
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Theorem 18
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In a plane, two lines parallel to a third line are parallel to each other.
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Theorem 19
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Parallel lines form equal corresponding angles.
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Corollary 1 to Theorem 19
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Parallel lines form equal alternate interior angles.
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Corollary 2 to Theorem 19
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Parallel lines form supplementary angles on the same side of the transversal.
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Corollary 3 to Theorem 19
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In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other.
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Theorem 20 (Triangle Angle Sum Theorem)
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The sum of the angles of a triangle is 180 degrees.
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Corollary 1 to Theorem 20
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If two angles of a triangle are equal to the angles of another, the third pair are equal.
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Corollary 2 to Theorem 20
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The acute angles of a right triangle are complementary.
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Corollary 3 to Theorem 20
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Each angle of an equilateral triangle is 60 degrees.
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Theorem 21
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An exterior angle of a triangle is equal to the sum of the remote interior angles.
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Theorem 22 (AAS)
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If two angles and a side opposite one of them in a triangle are equal to those of another, the two triangles are congruent.
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Theorem 23 (HL)
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If the hypotenuse and a leg of a right triangle are equal to those of another, the two triangles are congruent.
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Theorem 24
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The sum of the angles of a quadrilateral is 360 degrees.
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Corollary to Theorem 24
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A quadrilateral is equiangular iff it is a rectangle.
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Theorem 25
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The opposite sides and angles of a parallelogram are equal.
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Theorem 26
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The diagonals of a parallelogram bisect each other.
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Theorem 27
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A quadrilateral is a parallelogram if its opposite sides are equal.
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Theorem 28
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A quadrilateral is a parallelogram if its opposite angles are equal.
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Theorem 29
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A quadrilateral is a parallelogram if two opposites sides are both equal and parallel.
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Theorem 30
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A quadrilateral is a parallelogram if its diagonals bisect each other.
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Theorem 31
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All rectangles are parallelograms.
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Theorem 32
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All rhombuses are parallelograms.
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Theorem 33
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The diagonals of a rectangle are equal.
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Theorem 34
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The diagonals of a rhombus are perpendicular.
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Theorem 35
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The base angles of an isosceles trapezoid are equal.
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Theorem 36
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The diagonals of an isosceles trapezoid are equal.
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Theorem 37 (Triangle Midsegment Theorem)
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A midsegment of a triangle is parallel to the third side and half the length.
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Theorem 38
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The area of a right triangle is half the product of its legs.
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Theorem 39
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The area of a triangle is half the product of any base and corresponding altitude.
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Corollary to Theorem 39
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Triangles with equal bases and equal altitudes have equal areas.
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Theorem 40
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The area of a parallelogram is the product of any base and corresponding altitude.
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Theorem 41
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The area of a trapezoid is half the product of its altitude and the sum of its bases.
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Theorem 42 (Pythagorean Theorem)
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The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs.
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Theorem 43
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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, the triangle is a right triangle.
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Theorem 44 (Side-splitter Theorem)
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If a line parallel to one side of a triangle intersects the other two sides in different points, it divides the sides in the same ratio.
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Corollary to Theorem 44
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If a line parallel to one side of a triangle intersects the other two sides in different points, it cuts off segments proportional to the sides.
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Theorem 45
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If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
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Corollary to Theorem 45
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Two triangles similar to a third triangle are similar to each other.
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Theorem 46
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Corresponding altitudes of similar triangles have the same ratio as that of the corresponding sides.
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SAS Similarity
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If an angle of one triangle is equal to an angle of another triangle and the sides including these angles are proportional, the triangles are similar.
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SSS Similarity
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If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.
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Theorem 47
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The ratio of the perimeters of two similar polygons is equal to the ratio of the corresponding sides.
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Theorem 48
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The ratio of the areas of two similar polygons is equal to the square of the ratio of the corresponding sides.
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Theorem 49
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The altitude to the hypotenuse of a right triangle forms two triangles similar to it and to each other.
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Corollary one to Theorem 49
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The altitude to the hypotenuse of a right triangle is the geometric mean between the segments into which it divides the hypotenuse.
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Corollary two to Theorem 49
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Each leg of a right triangle is the geometric mean between the hypotenuse and its projection on the hypotenuse.
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Theorem 50 (Isosceles Right Triangle Theorem)
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In an isosceles right triangle, the hypotenuse is (square root) 2 times the length of a leg.
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Corollary to Theorem 50
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Each diagonal of a square is (square root) 2 times the length of one side.
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Theorem 51 (30-60 Right Triangle Theorem)
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In a 30-60 right triangle, the hypotenuse is twice the shorter leg and the longer leg is (square root)3 times the shorter leg.
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Corollary to Theorem 51
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An altitude of an equilateral triangle having side S is (square root)3/2(s) and its area is (square root)3/4(s^2).
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Theorem 52
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Two nonvertical lines are parallel iff their slopes are equal.
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Theorem 53
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Two nonvertical lines are perpendicular iff the product of their slopes is -1.
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Theorem 54 (Law of Sines)
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If the sides opposite angles A, B, and C in triangle ABC have lengths a, b, and c, then sinA/a = sinB/b = sinC/c.
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Theorem 55 (Law of Cosines)
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If sides opposite angles A, B, and C of triangle ABC have lengths of a, b, and c, then c^2 = a^2 + b^2 - (2ab)(cos C)
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Theorem 56
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If a line through the center of a circle is perpendicular to a chord, it also bisects the chord.
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Theorem 57
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If a line through the center of a circle bisects a chord that is not a diameter, it is also perpendicular to the chord.
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Theorem 58
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The perpendicular bisector of a chord of a circle contains the center of the circle.
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Theorem 59
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If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact.
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Theorem 60
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If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle.
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Theorem 61
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In a circle, equal chords have equal arcs.
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Theorem 62
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In a circle, equal arcs have equal chords.
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Theorem 63
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An inscribed angle is equal in measure to half its intercepted arc.
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Corollary 1 to Theorem 63
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Inscribed angles that intercept the same arc are equal.
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Corollary 2 to Theorem 63
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An angle inscribed in a semi-circle is a right angle.
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Theorem 64 (Secant angles)
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A secant angle whose vertex is inside a circle is equal in measure to half the sum of the arcs intercepted by it and its vertical angle.
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Theorem 65 (Secant angles)
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A secant angle whose vertex is outside a circle is equal to half the difference of its larger and smaller intercepted arcs.
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Theorem 66
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The tangent segments to a circle from an external point are equal.
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Theorem 67
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If two chords intersect in a circle, the product of the lengths of the segments is equal to the products of the lengths of the segments of the other chord.
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Theorem 68 (Start of concurrence chapter)
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Every triangle is cyclic.
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Corollary to Theorem 68
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The perpendicular bisectors of the sides of a triangle are concurrent.
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Theorem 69
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A quadrilateral is cyclic iff a pair of its opposite angles are supplementary.
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Theorem 70
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Every triangle has an incircle.
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Corollary to Theorem 70
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The angle bisectors of a triangle are concurrent.
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Theorem 71
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The medians of a triangle are concurrent.
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Theorem 72
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The lines containing the altitudes of a triangle are concurrent.
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Theorem 73 (Ceva's Theorem)
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Three cevians, AY, BZ, and CX, are concurrent iff (AX/XB)(BY/YC)(CZ/ZA) = 1.
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Theorem 74
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Every regular polygon is cyclic.
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Theorem 75 (Perimeter of a regular polygon)
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The perimeter of a regular polygon having n sides is 2Nr, in which N = n sin(180/n) and r is its radius.
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Theorem 76 (Area of a regular polygon)
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The area of a regular polygon having n sides is Mr^2, in which M = n(sin(180/n)cos(180/n)) and r is its radius.
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Theorem 77 (Circumference of a circle)
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If the radius of a circle is r, its circumference is 2(pi)r.
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Corollary to Theorem 77
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If the diameter of a circle is d, its circumference is (pi)d.
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Theorem 78 (Area of a circle)
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If the radius of a circle is r, its area is (pi)r^2.
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Theorem 79
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The length of a diagonal of a rectangular solid with dimensions l, w, and h is (squareroot)(l^2 + w^2 + h^2)
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Corollary to Theorem 79
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The length of a diagonal of a cube with length e is e(squareroot)3.
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Theorem 80 (Volume of a pyramid)
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The volume of any pyramid is 1/3Bh.
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Theorem 81 (Volume of a cylinder)
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The volume of a cylinder is (pi)r^2h.
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Theorem 82 (Volume of a cone)
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The volume of a cone is 1/3(pi)r^2h.
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Theorem 83 (Volume of a sphere)
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The volume of a sphere is 4/3(pi)r^3.
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Theorem 84 (Surface area of a sphere)
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The surface area of a sphere is 4(pi)r^2.
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Theorem 85
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If the ratio of a pair of corresponding dimensions of two similar solids is r, then the ratio of their surface area is r^2.
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Theorem 86
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If the ratio of a pair of corresponding dimensions of two similar solids is r, then the ratio of their volumes is r^3.
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Theorem 87
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The summit angles of a Saccheri quadrilateral are equal.
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Theorem 88
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The line segment connecting the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both of them.
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Theorem 89
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If the legs of a birectangular quadrilateral are unequal, the summit angles opposite them are unequal in the same order.
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Theorem 90
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If the summit angles of a birectangular quadrilateral are unequal, the legs opposite them are unequal in the same order.
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Lobachevskian Theorem 1
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The summit of a Saccheri quadrilateral is longer than its base.
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Lobachevskian Theorem 2
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A midsegment of a triangle is less than half as long as the third side.
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Lobachevskian Theorem 3
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The sum of the angles of a triangle is less than 180 degrees.
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Corollary to Lobachevskian Theorem 3
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The sum of the angles of a quadrilateral is less than 360 degrees.
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Lobachevskian Theorem 4
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If two triangles are similar, they must also be congruent.