Theorems

Navigation

Content

1 (SAS)
Two triangles are congruent if two sides and the included angle of one are equal respectively, only to two sides and the included angle of the other triangle.
2 (ASA)
Two triangles are congruent if a side and two adjacent angles of one triangle are equal respectivaly to a side and the two adjacent angles of another triangle.
3
In any isosceles triangle the angles opposite the equal side are equal
(Base angles of isosceles triangles are equal?)
4 (SSS)
Two triangles are congruent if the three sides of one triangle are equal respectively to the three sides of the other
5
An exterior angle of a triangle is greater than either opposite interior angles
6
If two angles of a triangle are equal, then sides opposite to those angles are equal
7
When two straight lines are cut by a transversal, if a pair of alternate interior angles are equal, then the two straight lines are parallel
8
If two parallel lines are cut by a transversal, then the alternate interior angles are equal
9
If two angles have their sides parallel right side to right side and left side parallel to left side, then their angles are equal
10
If two angles have their sides perpendicular right side to right side and left side to left side, then the angles are equal
11
The sum of the angles of a triangle equal one straight angle
12 (HL)
Two right triangles are congruent if the hypotenuse and leg of one triangle are equal respectively to the hypotenuse and one leg of the other triangle
13
If one side of a triangle is greater than a second side, then the angle opposite the greater side is greater than the angle opposite to the smaller side
14
If one angle of a triangle is greater than a second angle, then the side opposite to the greater angle is greater than the side opposite to the smaller angle
15
If two straight lines drawn from a point in a perpendicular are cut off equal segments from the foot of the perpendicular, then the segments are equal (?)
16
If two straight lines from a point in a perpendicular fashion to a given line are equal, then they cut off equal segmentsfrom the foot of the perpendicular
17
If two triangles have two sides of one are equal respectively to two sides of the other and the included angles are unequal, then the triangle with the greater included angle has a greater third side
18
If two triangles have two sides of one equal to two sides spective the other and the third side unequal the triangle with the greater third side has the greater included angle
19
Opposite sides of a parallelogram are equal, and the opposite angles are equal
20
Diagonals of a parallelogram bisect each other
21
If the opposite sides of a quadrilateral are equal, it is a parallelogram
22
If two sides of a quadrilateral are equal and parallel, it is a parallelogram
23
If diagonals of a quadrilateral bisect each other, it is a parallelogram
24
If three or more parallel divide one transversal into equal parts, they divide any of the transversal into equal parts (?)
25
A line joining the mid points of two sides of a triangle is parallel the third and is equal to 1/2 the third side
26
The sum of the angles of a polygon of n-sides is (n-2) straight angles (or (n-2)*180 degrees)
27-A
Every point in the perpendicualr bisector of a line segment is equidistant from the end of the line
27-B
Every point equidistant from the ends of a line segment lies on a perpendicular bisector of that line
28-A
Every point in the bisector of an angle is equidistant from the sides of that angle
28-B
Every point equidistant from the sides of an angle lies in the bisector of that angle
29
perpendicular bisectors of the sides of a triangle meet in a point which is equidistant from the three vertices
30
Bisectors of the angles of a triangle meet at a point which is equidistant from the three sides
31
Medians of a triangle are concurrent at a point which is 2/3 distant from each vertex to the mid point of the opposite side. (centroid?)
32-A
If in the same circle or in equivalent circles, two central angles are equal, then the arcs which they intercept are equal
32-B
If in the same circle or in equivalent circles, two arcs are equal
33-A
If in the same circle or in equivalent circles two arcs are equal, then the chords subtended by them are equal
33-B
If in the same circle or in equivalent circles two chords are equal, then the arcs subtended by them are equal
34-A
If in the same circle or in equivalent circles two chords are unequal, then the greater chord subtends the greater minor arc
34-B
If in the same circle or in equivalent circles two minor arcs are unequal, then the greater arc subtends the greater chord
35
A diameter or a radius perpendicular to a chord bisecs the chord and the arcs subtended by it
36-A
If in the same circle or in equivalent circles two chords are equal, then they are equidistant from the center
36-B
If in the same circle or in equivalent circles two chords are equidistant from the center, then they are equal
37
If in the same circle or in equivalent circles two chords are unequal, then the smaller chord is nearer the center
38
If in the same circle or in equivalent circles two chords are not equidistant from the center, then the chord nearer to the center is the smaller chord
39
A tangent to a circle is perpendicular to the radius or diameter at the point of tangency
40
Tangent drawn to a circle from an external point are equal
41
When two circle intersect each other, the line of centers is perpendicular bisector of the common chord
42
An angle inscribed in a circle is measured by 1/2 its intercepted arc
43
An angle formed by two chords intersecting within a circle is measured by 1/2 the sum of the arc intercepted between its sides and the arc intercepted between the sides of its vertical angle
44
An angle formed by a tangent and a chord drawn from the point of tangency is measured by 1/2 its intercepted arc
45
An angle formed by two secants, a secant and a tangent, or two tangents intersecting outside of a circle, is measured by 1/2 the difference of the intercepted arcs
46
Arcs of a circle intercepted between two parallel lines are equal
47
The locus of points equidistant from two given points is the perpendicular bisector of the line joining the two points
48
The locus of points within an angle equidistant from the sides is a line that bisects the angle
49
The locus of points equidistant from two parallel lines is aline parallel to each given lines and midway between them
50
The locus of points a given distance from a given line consists of two lines, one on either side of teh given line, both parallel to the given line and the given distance away from it
51
The locus of points on a given distance from a given point is the circle described with the given point as center and given distance as radius
52
The locus of centers of all circles tangent to a given line at a given point is perpendicular to the line at that point
53
If a linethrough two sides of a triangle parallel to the third side, then it divides the two sides proportionally
54
If a line divides two sides of a triangle proportionally, then it is parallel to the third side
55
The bisector of an interior angle of a triangle divides the opposite side internally into segments which have the same ratio as the other two sides
56
The bisector of an exterior angle of a triangle divides the opposite side externally into segments which have the same ratio as the other two sides
57 (AAA)
If two triangles have three angles of one triangle equal respectively to the three angles of the other, then the triangles are similar (~)
58 (SAS)
If two triangles have two pairs of sides proportional and the included angles equal respectively, then the two triangles are similar (~)
59 (SSS)
If two triangles have all three sides respectively proportional, then the triangles are similar (~)
60
If two parallel lines are cut by three or more transversals passing through a common point, then the corresponding segments of the parallel lines are proportional
61
If in a right triangle the perpendicular drawn from the vertex of the right angle to the hypotenuse, the two triangles formed are similar to the given triangle and to each other.
The perpendicular is a mean proportional between segments of the hypotenuse.
Each leg of the given triangle is a mean proportion between the hypotenuse and adjacent segment
AD / CD = CD / BD
AB / AC = AC / AD
AB / BC = BC / BD
62
c2 = a2 + b2
63
If two chords intersect within the circle, then the product of the segment of one chord = product of the segments of the other chord
64
If from a point outside of a circle, tangents and secants drawn to the circle, then the tangent is the mean proportional between the secant and its external segment
65
perimeters of two similar (~) p-gons are to each other as any two common sides
66
If two p-gons are ~ (similar), then they may be decomposed into the same [illegible - tpf?] triangles ~ to each other and similarly placed
67
If two p-gons are composed of the same number of triangles similar to each, then the p-gons are similar
68
The area of a rectangle = the product of its base and altitude
R = bh
69
The area of a parallelogram = the product of its base and altitude
P = bh
70
The area of a triangle = 1/2 the product of its base and altitude
T = 1/2 bh
71
The area of a trapezoid = 1/2 the product of its altitude and the sum of its bases
Tr = 1/2 h(b1 + b2)
72
If two triangles have an angle of one equal to the angle of the other, their areas are to each other as the products of the sides including the equal angles
73
The areas of two ~ triangles are to each other as the [squares?] of any two corresponding sides
74
The areas of two ~ p-gons are to each other as the [squares?] of any two corresponding sides
75
The square of the hypotenuse of a right triangle = the sum of the squares of the two legs
76
Equilateral polygons inscribed in a circle is regular
77
If a circle divides into any number of equal parts:
A) chords joining successive points of [division?] form a regular polygon inscribed in a circle
B) tangents drawn at points of [division?] form a regular polygon circumscribed about the circle
78
A circle may be circumscribed about any regular polygon and any circle may be inscribed in it
79
regular polygons of the same number of sides are ~
80
Perimeters of two regular polygons of the same number of sides have the same ratio as radii as apothems
81
T [triangle?] = 1/2 (perimeter)(apothem)
82
Circumferences of two circles are in the same ratio as their radii
83
The area of a circle = 1/2 (C * r)