Geometry - Theorem List

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Theorem 001 (Page 024)

If two angles are right angles, then they are congruent.

Theorem 002 (Page 024)

If two angles are straight angles, then they are congruent.

Theorem 003 (Page 046)

If a conditional statement is true, then the contrapositive of the statement is also true. (If p, then q ⇔If ~q, then ~p.)

Theorem 004 (Page 076)

If angles are supplementary to the same angle, then they are congruent.

Theorem 005 (Page 077)

If angles are supplementary to congruent angles, then they are congruent.

Theorem 006 (Page 077)

If angles are complementary to the same angle, then they are congruent.

Theorem 007 (Page 077)

If angles are complementary to congruent angles, then they are congruent.

Theorem 008 (Page 082)

If a segment is added to two congruent segments, the sums are congruent. (Addition Property)

Theorem 009 (Page 083)

If an angle is added to two congruent angles, the sums are congruent. (Addition Property)

Theorem 010 (Page 083)

If congruent segments are added to congruent segments, the sums are congruent. (Addition Property)

Theorem 011 (Page 083)

If congruent angles are added to congruent angles, the sums are congruent. (Addition Property)

Theorem 012 (Page 084)

If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property)

Theorem 013 (Page 084)

If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property)

Theorem 014 (Page 089)

If segments (or angles) are congruent, their like multiples are congruent. (Multiplication Property)

Theorem 015 (Page 090)

If segments (or angles) are congruent, their like divisions are congruent. (Division Property)

Theorem 016 (Page 095)

If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. (Transitive Property)

Theorem 017 (Page 095)

If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other. (Transitive Property)

Theorem 018 (Page 101)

Vertical angles are congruent.

Theorem 019 (Page 126)

All radii of a circle are congruent.

Theorem 020 (Page 148)

If two sides of a triangle are congruent, the angles opposite the sides are congruent. (If , then )

Theorem 021 (Page 149)

If two angles of a triangle are congruent, then sides opposite the angles are congruent. (If , then )

Theorem 022 (Page 171)

If A=(x1,y1) and B=(x2,y2), then the midpoint M=(xm,ym) of (AB) ̅ can be found by using the midpoint formula: M=(x_m,y_m )=((x_1+x_2)/2,(y_1+y_2)/2)

Theorem 023 (Page 180)

If two angles are both supplementary and congruent, then they are right angles.

Theorem 024 (Page 185)

If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

Theorem 025 (Page 195)

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.

Theorem 026 (Page 200)

If two non-vertical lines are parallel, then their slopes are equal.

Theorem 027 (Page 200)

If the slopes of two non-vertical lines are equal, then the lines are parallel.

Theorem 028 (Page 200)

If two lines are perpendicular and neither is vertical, each line's slope is the opposite reciprocal of the other's.

Theorem 029 (Page 200)

If a line's slope is the opposite reciprocal of another line's slope, the two lines are perpendicular.

Theorem 030 (Page 216)

The measure of an exterior angle of a triangle is greater than the measure of either remove interior angle.

Theorem 031 (Page 217)

If two lines are cut by a traversal such that two alternate interior angles are congruent, the lines are parallel. (Alt. int. ⦞ ≌⇒‖lines)

Theorem 032 (Page 217)

If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel. (Alt. ext. ⦞ ≌⇒‖ lines)

Theorem 033 (Page 217)

If two lines are cut by a traversal such that two corresponding angles are congruent, the lines are parallel. (Corr. ⦞ ≌⇒‖lines)

Theorem 034 (Page 218)

If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are parallel.

Theorem 035 (Page 218)

If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are parallel.

Theorem 036 (Page 218)

If two coplanar lines are perpendicular to a third line, they are parallel.

Theorem 037 (Page 225)

If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent. (‖ lines ⇒ alt. int. ⦞ ≌)

Theorem 038 (Page 225)

If two parallel lines are cut by a transversal, then any pair of the angles are either congruent or supplementary.

Theorem 039 (Page 226)

If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent. (‖ lines ⇒ alt. ext. ⦞ ≌)

Theorem 040 (Page 226)

If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. (‖ lines ⇒ corr. ⦞ ≌)

Theorem 041 (Page 226)

If two parallel IF two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary.

Theorem 042 (Page 226)

If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the transversal are supplementary.

Theorem 043 (Page 227)

In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.

Theorem 044 (Page 227)

If two lines are parallel to a third line, they are parallel to each other. (Transitive Property of Parallel Lines)

Theorem 045 (Page 271)

A line and a point not on the line determine a plane.

Theorem 046 (Page 271)

Two intersecting lines determine a plane.

Theorem 047 (Page 271)

Two parallel lines determine a plane.

Theorem 048 (Page 277)

If a line is perpendicular to two distinct lines that lie in a plane and that pass through its foot, then it is perpendicular to the plane.

Theorem 049 (Page 283)

If a plane intersects two parallel planes, the lines of intersection are parallel.

Theorem 050 (Page 295)

The sum of the measures of the three angles of a triangle is 180.

Theorem 051 (Page 296)

The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Theorem 052 (Page 296)

A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-half the length of the third side. (Midline Theorem)

Theorem 053 (Page 302)

If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (No-Choice Theorem)

Theorem 054 (Page 302)

If there exists a correspondence between the vertices of two triangles such that two angles and a non-included side of one are congruent to the corresponding parts of the other, then the triangles are congruent. (AAS)

Theorem 055 (Page 308)

The sum Si of the measures of the angles of a polygon with n sides is given by the formula Si = (n - 2)180.

Theorem 056 (Page 308)

If one exterior angle is taken at each vertex, the sum Se of the measures of the exterior angles of a polygon is given by the formula Se = 360.

Theorem 057 (Page 308)

The number d of diagonals that can be drawn in a polygon of n sides is given by the formula d=(n(n-3))/2.

Theorem 058 (Page 315)

The measure E of each exterior angle of an equiangular polygon of n sides is given by the formula E=360/n.

Theorem 059 (Page 327)

In a proportion, the product of the means is equal to the product of the extremes. (Means-Extremes Products Theorem)

Theorem 060 (Page 327)

If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion. (Means-Extremes Ratio Theorem)

Theorem 061 (Page 334)

The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.

Theorem 062 (Page 339)

If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar. (AA)

Theorem 063 (Page 340)

If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of corresponding sides equal, then the triangles are similar. (SSS~)

Theorem 064 (Page 340)

If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar. (SAS~)

Theorem 065 (Page 351)

If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. (Side-Splitter Theorem)

Theorem 066(Page 351)

If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

Theorem 067 (Page 352)

If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem)

Theorem 068 (Page 378)

If an altitude is drawn to the hypotenuse of a right triangle, then
The two triangles formed are similar to the given right triangle and to each other
The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse
Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg (i.e., the projection of that leg on the hypotenuse)

Theorem 069 (Page 384)

The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs. (Pythagorean Theorem)

Theorem 070 (Page 385)

If the square of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle.

Theorem 071 (Page 393)

If P=(x1,y1) and Q=(x2,y2), are any two points, then the distance between them can be found with the formula PQ= √((x_2-x_1 )+(y_2-y_1)) or PQ=√(〖(△x)〗^2+〖(△y)〗^2 )

Theorem 072 (Page 405)

In a triangle whose angles have the measures 30, 60, and 90, the lengths of the sides opposite these angles can be represented by x, x√3, and 2x respectively. (30º-60º-90º Triangle Theorem)

Theorem 073 (Page 406)

In a triangle whose angles have the measures 45, 45, and 90, the lengths of the sides opposite these angles can be represented by x, x, and x√2 respectively. (45º-45º-90º Triangle Theorem)

Theorem 074 (Page 441)

If a radius is perpendicular to a chord, then it bisects the chord.

Theorem 075 (Page 441)

If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord.

Theorem 076 (Page 441)

The perpendicular bisector of a chord passes through the center of a circle.

Theorem 077 (Page 446)

If two chords of a circle are equidistant from the center, then they are congruent.

Theorem 078 (Page 446)

If two chords of a circle are congruent, then they are equidistant from the center of the circle.

Theorem 079 (Page 453)

If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent.

Theorem 080 (Page 453)

If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.

Theorem 081 (Page 453)

If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.

Theorem 082 (Page 453)

If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.

Theorem 083 (Page 453)

If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.

Theorem 084 (Page 453)

If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are congruent.

Theorem 085 (Page 460)

If two tangent segments are drawn to a circle from an external point, then those segments are congruent. (Two-Tangent Theorem)

Theorem 086 (Page 469)

The measure of an inscribed angle or a tangent-chord angle (vertex on a circle) is one-half the measure of its intercepted arc.

Theorem 087 (Page 470)

The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle.

Theorem 088 (Page 471)

The measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle (vertex outside a circle) is one-half the difference of the measures of the intercepted arcs.

Theorem 089 (Page 479)

If two inscribed or tangent-chord angles intercept the same arc, then they are congruent.

Theorem 090 (Page 479)

If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent.

Theorem 091 (Page 480)

An angle inscribed in a semicircle is a right angle.

Theorem 092 (Page 480)

The sum of the measures of a tangent-tangent angle and its minor arc is 180.

Theorem 093 (Page 487)

If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.

Theorem 094 (Page 488)

If a parallelogram is inscribed is a circle, it must be a rectangle.

Theorem 095 (Page 493)

If two chords of a circle intersect inside the circle, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord. (Chord-Chord Power Theorem)

Theorem 096 (Page 493)

If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the measure of the tangent segment is equal to the product of the measures of the entire secant segment and its external part. (Tangent-Secant Power Theorem)

Theorem 097 (Page 494)

If two secant segments are drawn from an external point to a circle, then the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part. (Secant-Secant-Power Theorem)

Theorem 098 (Page 500)

The length of an arc is equal to the circumference of its circle times the fractional part of the circle determined by the arc.

Theorem 099 (Page 512)

The area of a square is equal to the square of a side.

Theorem 100 (Page 516)

The area of a parallelogram is equal to the product of the base and the height.

Theorem 101 (Page 517)

The area of a triangle is equal to one-half the product of a base and the height (or altitude) for that base.

Theorem 102 (Page 523)

The area of a trapezoid equals one-half the product of the height and the sum of the bases.

Theorem 103 (Page 524)

The measure of the median of a trapezoid equals the average of the measures of the bases.

Theorem 104 (Page 524)

The area of a trapezoid is the product of the median and the height.

Theorem 105 (Page 528)

The area of a kite equals half the product of its diagonals.

Theorem 106 (Page 531)

The area of an equilateral triangle equals the product of one-fourth the square of a side and the square root of 3.

Theorem 107 (Page 532)

The area of a regular polygon equals one-half the product of the apothem and the perimeter.

Theorem 108 (Page 537)

The area of a sector of a circle is equal to the area of a circle times the fractional part of the circle determined by the sector's arc.

Theorem 109 (Page 544)

If two figures are similar, then the ratio of their area equals the square of the ratio of corresponding segments. (Similar-Figures Theorem)

Theorem 110 (Page 546)

A median of a triangle divides the triangle into two triangles with equal areas.

Theorem 111 (Page 550)

Area of a triangle = √(s(s-a)(s-b)(s-c)) where a, b, and c are the lengths of the triangle and s = semi perimeter = (a+b+c)/2. (Hero's formula)

Theorem 112 (Page 550)

Area of a cyclic quadrilateral = √(s(s-a)(s-b)(s-c)(s-d)) where a, b, c, and d are the sides of the quadrilateral and s = semi perimeter = (a+b+c+d)/2. (Brahmagupta's formula)

Theorem 113 (Page 571)

The lateral area of a cylinder is equal to the product of the height and the circumference of the base.

Theorem 114 (Page 571)

The lateral area of a cone is equal to one-half the product of the slant height and the circumference of the base.

Theorem 115 (Page 576)

The volume of a right rectangular prism is equal to the product of the height and the area of the base.

Theorem 116 (Page 576)

The volume of any prism is equal to the product of the height and the area of the base.

Theorem 117 (Page 577)

The volume of a cylinder is equal to the product of the figures cross-sectional area and its height.

Theorem 118 (Page 577)

The volume of a prism or a cylinder is equal to the product of the figure's cross sectional area and its height.

Theorem 119 (Page 583)

The volume of a pyramid is equal to one third of the product of the height and the area of the base.

Theorem 120 (Page 584)

The volume of a cone is equal to one third of the product of the height and the area of the base.

Theorem 121 (Page 584)

In a pyramid or a cone, the ratio of the area of a cross section to the area of the base equals the square of the ratio of the figures' respective distances from the vertex.

Theorem 122 (Page 589)

The volume of a sphere is equal to four third of the product of π and the cube of the radius.

Theorem 123 (Page 610)

The y-form, or slope-intercept form, of the equation of a nonvertical line is y = mx+b, where b is the y-intercept of the line and m is the slope of the line.

Theorem 124 (Page 611)

The formula for an equation of a horizontal line is y = b, where b is the y-coordinate of every point on the line.

Theorem 125 (Page 612)

The formula for the equation of a vertical line is x = a, where a is the x-coordinate of every point on the line.

Theorem 126 (Page 626)

If P = (x1, y1 ,z1) and Q = (x2, y2 ,z2) are any two points, then the distance between them can be found with the formula PQ = √(〖(x_2-x_1)〗^2 〖+(y_2-y_1)〗^2+〖(z_2-z_1)〗^2 )

Theorem 127 (Page 633)

The equation of a circle whole center is (h, k) and whose radius is r is 〖(x-h)〗^2+〖(y-k)〗^2=r^2

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