Plane, Solid, & Coordinate GEOMETRY
**Definitions, properties, postulates, theorems, and corollaries provide the
reasoning for making logical statements in proofs.
Fundamentals
1. Post: Segment Addition – Point Q is between P and R iff PQ + QR = PR.
2. Post: Angle Addition – Point R is in the interior of PQS iff mPQR  mRQS  mPQS .
3. Thm: Midpoint Theorem – If M is the midpoint of AB , then AM = MB = ½ AB.
Logic
 Algebra Properties – Reflexive, Symmetric, Transitive, Substitution, Distribution, Properties of
Equality (addition, division, subtraction, multiplication)
4. Post: Through any 2 points in space, there is exactly 1 line.
5. Post: Through any 3 noncollinear points, there is exactly 1 plane.
6. Post: A line contains at least 2 points.
7. Post: A plane contains at least 3 noncollinear points.
8. Post: If 2 points lie in a plane, then the entire line containing them lies in that plane.
9. Post: If 2 planes intersect, then the intersection is a line.
10. Thm: Congruence of segments is reflexive, symmetric, and transitive.
11. Thm: Congruence of angles is reflexive symmetric, and transitive
12. Thm: (Supplement Theorem) If 2 angles form a linear pair, then they are supplementary.
13. Thm: Two angles that are supplementary to the same angle are congruent.
14. Thm: Two angles that are complementary to the same angle are congruent.
15. Thm: All right angles are congruent
16. Thm: Vertical angles are congruent
17. Thm: Perpendicular lines intersect to form four right angles.
Lines & Angles
18. Post:
19. Thm:
20. Thm:
21. Thm:
22. Thm:
23. Post:
24. Thm:
25. Thm:
26. Thm:
27. Thm:
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
If two parallel lines are cut by a transversal, then alternate exterior angles is congruent.
If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.
If two parallel lines are cut by a transversal, then consecutive exterior angles are supplementary.
If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the
lines are parallel.
If two lines in a plane are cut by a transversal so that a pair alternate exterior angles are
congruent, then the lines are parallel.
If two lines in a plane are cut by a transversal so that a pair of alternate interior angles are
congruent, then the lines are parallel.
If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles are
supplementary, then the lines are parallel.
If two lines in a plane are cut by a transversal so that a pair of consecutive exterior angles are
supplementary, then the lines are parallel.
28. Thm: If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line.
29. Thm: In a plane, if two lines are perpendicular to the same line, then they are parallel.
30. Post: If there is a line and a point not on the line, then there exists exactly one line through the point
that is parallel to the given line.
31. Post: Two [nonvertical] lines are parallel iff they have the same slope.
32. Post: Two [nonvertical] lines are perpendicular iff their slopes are opposite reciprocals (product = –1).
Congruent Triangles
33. Thm: Angle Sum Theorem – The sum of the three angle measures in a triangle is 180°.
34. Thm: Third Angle Theorem – If two angles of one triangle are congruent to two angles of another
triangle, then the third angles of the triangles are also congruent.
35. Thm: Exterior Angle Theorem – The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two remote (opposite) interior angles.
36. Corollary – There can be at most one right or obtuse angle in a triangle.
37. Corollary – The acute angles of a right triangle are complementary.
38. Thm: Congruence of triangles is reflexive, symmetric, and transitive.
39. Post: (SSS) - If the sides of one triangle are congruent to the sides of a second triangle, then the triangles
are congruent.
40. Post: (SAS) – If two sides and the included angle of one triangle are congruent to two sides and the
included angled of another triangles, then the triangles are congruent.
41. Post: (ASA) - If two angles and the included side of one triangle are congruent to two angles and the
included side of another triangle, then the triangles are congruent.
42. Thm: (AAS) - If two angles and the non-included side of one triangle are congruent to the
corresponding two angles and side of a second triangle, then the two triangles are congruent.
43. Thm: Isosceles Triangle Theorem – If two sides of a triangle are congruent, then the angles opposite
those sides are congruent.
44. Thm: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
45. Corollary: A triangle is equilateral iff it is equiangular.
46. Corollary: Each angle of an equilateral triangle measures 60°.
Segments & Inequalities in Triangles
A point on the perpendicular bisector of a segment is equidistant from the segment’s endpoints.
A point that is equidistant from the endpoints of a segment lies on the perpendicular bisector.
A point on an angle bisector is equidistant from the sides of the angle.
Any point on the interior of an angle that is equidistant from the sides of the angle lies on the
angle bisector.
51. Thm: (LL) - If the legs of one right triangle are congruent to the corresponding legs of another right
triangle, then the triangles are congruent.
52. Thm: (HA) - If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse
and corresponding acute angle of another right triangle, then the triangles are congruent.
53. Thm: (LA) - If one leg and and an acute angle of one right triangle are congruent to the corresponding
leg and acute angle of another right triangle, then the triangles are congruent.
54. Post: (HL) – If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and
corresponding leg of another right triangle, then the triangles are congruent.
55. Thm: Exterior Angle Inequality – The measure of an exterior angle is greater than each of the measures
of the remote (opposite) interior angles.
56. Thm: If one side of a triangle is longer than another, then the angle opposite the longer side has a larger
measure than the angle opposite the shorter side.
47. Thm:
48. Thm:
49. Thm:
50. Thm:
57. Thm: If one angle of a triangle is larger than another angle, then the side opposite the larger angle is
longer than the side opposite the smaller angle.
58. Thm: The shortest distance from a point to a line is a perpendicular segment from the point to the line.
59. Corollary: The shortest distance from a point to a plane is the perpendicular from the point to the plane.
60. Thm: Triangle Inequality Theorem – The sum of any two sides of a triangle is greater than the third.
61. Thm: (SAS Inequality) – If two sides of one triangle are congruent to two sides of another triangle, and
included angle in one triangle is greater than included angle in other triangle, then third side of
first triangle is longer than the third side in second triangle.
62. Thm: (SSS Inequality) – If two sides of one triangle are congruent to two sides of another triangle, and
the third side in one triangle is longer than the third side of the other triangle, then the angle
between the pair of congruent sides in the first triangle is greater than the corresponding angle in
the second triangle.
Similar Triangles
63. Thm: (AA Similarity) – If two angles of one triangle are congruent to two angles of another, then the
triangles are similar.
64. Thm: (SSS Similarity) – If the measures of the corresponding sides of two triangles are proportional,
then the triangles are similar.
65. Thm: (SAS Similarity) – If the measures of two sides in one triangle are proportional to the measures
of two corresponding sides in another triangle and the angles between those sides are congruent,
then the triangles are similar.
66. Thm: Similarity is reflexive, symmetric, and transitive.
67. Thm: (Triangle Proportionality) – If a line is parallel to one side of a triangle and intersects the other
two sides in two distinct points, then it separates the sides into segments of proportional lengths.
68. Thm: If a line intersects two sides of a triangle and cuts the sides into proportional parts, then the line
is parallel to the third side of the triangle.
69. Corollary: If three or more parallel lines intersect two transversals, then they cut off the transversals
proportionally.
70. Corollary: If parallel lines cut off congruent segments on one transversal, then they cut
off congruent segments on every transversal.
71. Thm: A segment whose endpoints are the midpoints of two sides of a triangle (midline) is parallel to
the 3rd side and is also ½ the length of the 3rd side.
72. Thm: (Proportional Perimeters) – If two triangles are similar, then the perimeters are proportional to
any pair of corresponding sides…the perimeters have the same scale factor as the sides do.
73. Thm: If two triangles are similar, then the lengths of all corresponding medians, altitudes, and angle
bisectors are proportional to any pair of corresponding sides…they have the same scale factor.
74. Thm: (Angle Bisector Theorem) – An angle bisector in a triangle separates the opposite side into
segments that have the same ratio as the other two sides.
Quadrilaterals
75. Thm:
76. Thm:
77. Thm:
78. Thm:
79. Thm:
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Consecutive angles in a parallelogram are supplementary.
The diagonals of a parallelogram bisect each other.
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
80. Thm: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
81. Thm: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
82. Thm: If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the
quadrilateral is a parallelogram.
83. Thm: If a parallelogram is a rectangle, then its diagonals are congruent.
84. Thm: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
85. Thm: The diagonals of a rhombus are perpendicular.
86. Thm: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
87. Thm: Each diagonal of a rhombus bisects a pair of opposite angles.
88. Thm: Both pairs of base angles of an isosceles trapezoid are congruent.
89. Thm: The diagonals of an isosceles trapezoid are congruent.
90. Thm: The median of a trapezoid is parallel to the bases, and its measure is half of the sum of the bases.
Right Triangles & Trigonometry
91. Thm: If the altitude is drawn to the hypotenuse in a right triangle, then the two triangles formed are
similar to each other as well as similar to the original right triangle.
92. Thm: The measure (length) of the altitude that is drawn to the hypotenuse of a right triangle is the
geometric mean between the measure of the two new segments formed on the hypotenuse.
93. Thm: If the altitude is drawn to the hypotenuse of a right triangle, then the measure of the leg of the
triangle is the geometric mean between the measures of the hypotenuse and the segment of the
hypotenuse adjacent to the leg.
94. Thm: (Pythagorean Theorem) – In a right triangle, the sum of the squares of the measures of the legs
equals the square of the measure of the hypotenuse
95. Thm: If the measures of the sides of a triangle satisfy the Pythagorean Equation, then the triangle is a
right triangle.
96. Thm: In a 45°-45°-90° isosceles right triangle, the hypotenuse is 2 times as long as a leg.
97. Thm: In a 30°-60°-90° right triangle, the hypotenuse is twice as long as the shorter leg, and the longer
leg is 3 times as long as the shorter leg.
Circles
98. Post: (Arc Addition) – The measure of an arc formed by two adjacent arcs is the sum of the measures of
the two arcs. That is, if Q is a point on PR , then mPQ  mQR  PR .
99. Thm: In a circle, two minor arcs are congruent iff their corresponding chords are congruent.
100. Thm: In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.
101. Thm: In a circle, two chords are congruent iff they are equidistant from the center.
102. Thm: If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of
the intercepted arc.
103. Thm: If two inscribed angles of a circle intercept congruent arcs or the same arc, then the angles are
congruent.
104. Thm: If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.
105. Thm: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
106. Thm: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of
tangency.
107. Thm: In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the circle, then the
line is a tangent of the circle.
108. Thm: If two segments from the same exterior point are tangent to a circle, then they are congruent.
109. Thm: If a secant and a tangent intersect at the point of tangency, then the measure of each angle
formed is one-half the measure of its intercepted arc.
110. Thm: If two secants intersect in the interior of a circle, then the measure of an angle formed is onehalf the sum of the measure of the arcs intercepted by the angle and its vertical angle.
111. Thm: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then
the measure of the angle formed is one-half the positive difference of the measure of the intercepted
arcs.
112. Thm: If two chords intersect in a circle, then the products of the measures of the segments of the
chords are equal.
113. Thm: If two secant segments are drawn to a circle from an exterior point, then the product of the
measures of one secant segment and its external secant segment is equal to the product of the measure
of the other secant segment and its external secant segment.
114. Thm: If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the
square of the measures of the tangent segment is equal to the product of the measures of the secant
segment and its external secant segment.
Polygons & Area
115. Thm: (Interior Angle Sum) – If a convex polygon has n sides and S is the sum of the measures of all
the interior angles, then S = 180(n – 2).
116. Thm: (Exterior Angle Sum) – If a polygon is convex, then the sum of the measures of all the exterior
angles, one at each vertex, is 360°.
117. Post: The area of a region is the sum of the areas of all of its non-overlapping parts.
118. Post: Congruent figures have equal areas.
119. Post: (Length Probability) – If a point on AB is chosen at random and C is between A and B, then the
probability that the point is on AC is AC / AB.
120. Post: (Area Probability) – If a point in region A is chosen at random, then the probability that the point
area ( B )
is in region B, which is in the interior of region A, is
.
area ( A)
Surface Area & Volume
121. Thm: If two solids are similar with a scale factor of a:b, then the surface areas have a ratio of
a2:b2 and the volumes have a ratio of a3:b3.
More Coordinate Geometry
122. Thm: (Slope-Intercept Form) – If the equation of a line is written in the form y = mx + b, then m is
the slope of the line and b is the y-intercept.
123. Thm: Given two points A(x1, y1, z1) and B(x2, y2, z2) in space, the distance between A and B is given
by the equation: AB  ( x2  x1 )2  ( y2  y1 )2  ( z2  z1 )2