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Theorem 1

answer

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

answer

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

answer

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)

answer

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

answer

The sum of the angles of a quadrilateral is 360 degrees.

question

Corollary to Theorem 24

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A quadrilateral is equiangular iff it is a rectangle.

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Theorem 25

answer

The opposite sides and angles of a parallelogram are equal.

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Theorem 26

answer

The diagonals of a parallelogram bisect each other.

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Theorem 27

answer

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

answer

A quadrilateral is a parallelogram if two opposites sides are both equal and parallel.

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Theorem 30

answer

A quadrilateral is a parallelogram if its diagonals bisect each other.

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Theorem 31

answer

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

answer

The diagonals of a rhombus are perpendicular.

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Theorem 35

answer

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

answer

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

answer

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.

question

SSS Similarity

answer

If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar.

question

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

answer

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

answer

The altitude to the hypotenuse of a right triangle forms two triangles similar to it and to each other.

question

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.

question

Corollary two to Theorem 49

answer

Each leg of a right triangle is the geometric mean between the hypotenuse and its projection on the hypotenuse.

question

Theorem 50 (Isosceles Right Triangle Theorem)

answer

In an isosceles right triangle, the hypotenuse is (square root) 2 times the length of a leg.

question

Corollary to Theorem 50

answer

Each diagonal of a square is (square root) 2 times the length of one side.

question

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.

question

Corollary to Theorem 51

answer

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

answer

Two nonvertical lines are parallel iff their slopes are equal.

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Theorem 53

answer

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

answer

If a line through the center of a circle is perpendicular to a chord, it also bisects the chord.

question

Theorem 57

answer

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

answer

The perpendicular bisector of a chord of a circle contains the center of the circle.

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Theorem 59

answer

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

answer

If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle.

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Theorem 61

answer

In a circle, equal chords have equal arcs.

question

Theorem 62

answer

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

answer

Inscribed angles that intercept the same arc are equal.

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Corollary 2 to Theorem 63

answer

An angle inscribed in a semi-circle is a right angle.

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Theorem 64 (Secant angles)

answer

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)

answer

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

answer

The tangent segments to a circle from an external point are equal.

question

Theorem 67

answer

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)

answer

Every triangle is cyclic.

question

Corollary to Theorem 68

answer

The perpendicular bisectors of the sides of a triangle are concurrent.

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Theorem 69

answer

A quadrilateral is cyclic iff a pair of its opposite angles are supplementary.

question

Theorem 70

answer

Every triangle has an incircle.

question

Corollary to Theorem 70

answer

The angle bisectors of a triangle are concurrent.

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Theorem 71

answer

The medians of a triangle are concurrent.

question

Theorem 72

answer

The lines containing the altitudes of a triangle are concurrent.

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Theorem 73 (Ceva's Theorem)

answer

Three cevians, AY, BZ, and CX, are concurrent iff (AX/XB)(BY/YC)(CZ/ZA) = 1.

question

Theorem 74

answer

Every regular polygon is cyclic.

question

Theorem 75 (Perimeter of a regular polygon)

answer

The perimeter of a regular polygon having n sides is 2Nr, in which N = n sin(180/n) and r is its radius.

question

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.

question

Theorem 77 (Circumference of a circle)

answer

If the radius of a circle is r, its circumference is 2(pi)r.

question

Corollary to Theorem 77

answer

If the diameter of a circle is d, its circumference is (pi)d.

question

Theorem 78 (Area of a circle)

answer

If the radius of a circle is r, its area is (pi)r^2.

question

Theorem 79

answer

The length of a diagonal of a rectangular solid with dimensions l, w, and h is (squareroot)(l^2 + w^2 + h^2)

question

Corollary to Theorem 79

answer

The length of a diagonal of a cube with length e is e(squareroot)3.

question

Theorem 80 (Volume of a pyramid)

answer

The volume of any pyramid is 1/3Bh.

question

Theorem 81 (Volume of a cylinder)

answer

The volume of a cylinder is (pi)r^2h.

question

Theorem 82 (Volume of a cone)

answer

The volume of a cone is 1/3(pi)r^2h.

question

Theorem 83 (Volume of a sphere)

answer

The volume of a sphere is 4/3(pi)r^3.

question

Theorem 84 (Surface area of a sphere)

answer

The surface area of a sphere is 4(pi)r^2.

question

Theorem 85

answer

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.

question

Theorem 86

answer

If the ratio of a pair of corresponding dimensions of two similar solids is r, then the ratio of their volumes is r^3.

question

Theorem 87

answer

The summit angles of a Saccheri quadrilateral are equal.

question

Theorem 88

answer

The line segment connecting the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both of them.

question

Theorem 89

answer

If the legs of a birectangular quadrilateral are unequal, the summit angles opposite them are unequal in the same order.

question

Theorem 90

answer

If the summit angles of a birectangular quadrilateral are unequal, the legs opposite them are unequal in the same order.

question

Lobachevskian Theorem 1

answer

The summit of a Saccheri quadrilateral is longer than its base.

question

Lobachevskian Theorem 2

answer

A midsegment of a triangle is less than half as long as the third side.

question

Lobachevskian Theorem 3

answer

The sum of the angles of a triangle is less than 180 degrees.

question

Corollary to Lobachevskian Theorem 3

answer

The sum of the angles of a quadrilateral is less than 360 degrees.

question

Lobachevskian Theorem 4

answer

If two triangles are similar, they must also be congruent.

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