Plane, Solid, & Coordinate GEOMETRY
**Definitions, properties, postulates, theorems, and corollaries provide the
REASONS for our 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: If two angles are supplementary to the same angle, then the supplements are congruent.
13. Thm: If two angles are complementary to the same angle, then the complements are congruent.
14. Thm: All right angles are congruent
15. Thm: Vertical angles are congruent
16. Thm: Perpendicular lines intersect to form four right angles.
Lines & Angles
17. Post:
18. Thm:
19. Thm:
20. Thm:
21. Thm:
22. Post:
23. Thm:
24. Thm:
25. Thm:
26. 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 coplanar lines are cut by a transversal and corresponding angles are congruent, then
the lines are parallel.
If coplanar lines are cut by a transversal and alternate interior angles are congruent, then
the lines are parallel.
If coplanar lines are cut by a transversal and alternate exterior angles are congruent, then
the lines are parallel.
If coplanar lines are cut by a transversal and consecutive interior angles are supplementary, then
the lines are parallel.
If coplanar lines are cut by a transversal and consecutive exterior angles are supplementary, then
the lines are parallel.
27. Thm: If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
28. Thm: In a plane, if two lines are perpendicular to the same line, then they are parallel.
29. 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.
30. Post: Two [nonvertical] lines are parallel iff they have the same slope.
31. Post: Two [nonvertical] lines are perpendicular iff their slopes are opposite reciprocals (product = –1).
Triangles (the Basics, Segments, & Inequalities)
32. Thm: Angle Sum Theorem – The sum of the three angle measures in a triangle is 180°.
33. Corollary – There can be at most one right or obtuse angle in a triangle.
34. Corollary – The acute angles of a right triangle are complementary.
35. Thm: Exterior Angle Theorem – The measure of an exterior angle is equal to the sum of the remote, or
“far away”interior angles.
36. Thm: Exterior Inequality Theorem – The measure of an exterior angle is greater than either of the
remote interior angles.
37. Thm: A point is on the perpendicular bisector of a segment iff it is equidistant from the segment’s
endpoints.
38. Thm: A point is on an angle bisector iff it is equidistant from the sides of the angle.
39. Thm: Triangle Inequality Theorem – The sum of any two sides of a triangle is greater than the third.
40. 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.
41. 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.
42. Thm: The shortest distance from a point to a line is a perpendicular segment from the point to the line.
43. Corollary: The shortest distance from a point to a plane is the perpendicular from the point to the plane.
44. Thm: Isosceles Triangle Theorem – If two sides of a triangle are congruent, then the angles opposite
those sides are congruent.
45. Thm: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
46. Corollary: A triangle is equilateral iff it is equiangular.
47. Corollary: Each angle of an equilateral triangle measures 60°.
Congruent Triangles
48. 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.
49. Thm: Congruence of triangles is reflexive, symmetric, and transitive.
50. Post: (SSS) - If three sides of one triangle are congruent to three sides of another triangle, then the
triangles are congruent.
51. Post: (SAS) – If two sides and the included angle of one triangle are congruent to two sides and the
included angled of another triangle, then the triangles are congruent.
52. 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.
53. 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.
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
leg of another right triangle, then the triangles are congruent.
Similar Triangles
55. Thm: (AA Similarity) – If two angles of one triangle are congruent to two angles of another, then the
triangles are similar. **because of the Third Angle Theorem, this is actually AAA**
56. Thm: (SSS Similarity) – If the measures of all corresponding sides in two triangles are proportional,
then the triangles are similar.
57. Thm: (SAS Similarity) – If the measures of two corresponding sides in two triangles are proportional
and the included angles are congruent, then the triangles are similar.
58. Thm: Similarity is reflexive, symmetric, and transitive.
59. Thm: (Triangle Proportionality) – A line is parallel to one side of a triangle iff it divides the other two
sides into proportional parts.
60. Corollary: If three or more parallel lines intersect two transversals, then they cut the transversals into
proportional parts.
61. Corollary: If parallel lines cut off congruent segments on one transversal, then they cut
off congruent segments on every transversal.
62. Thm: If a segment connects the midpoints of two sides of a triangle (midline), then it is parallel to
the 3rd side and is also ½ the length of the 3rd side.
63. Thm: (Proportional Perimeters) – If two triangles are similar, then the perimeters are proportional to
any pair of corresponding side (same scale factor).
64. 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 (same scale factor).
65. Thm: (Angle Bisector Theorem) – An angle bisector in a triangle divides the opposite side into two
segments that are proportional to the two sides of the angle.
Quadrilaterals
66. Thm:
67. Thm:
68. Thm:
69. Thm:
70. Thm:
71. Thm:
72. Thm:
73. Thm:
74. Thm:
75. Thm:
76. Thm:
77. Thm:
78. Thm:
79. Thm:
80. Thm:
81. 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 it is a parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram.
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
If one pair of opposite sides of a quadrilateral is both parallel and congruent, then it is
a parallelogram.
If a parallelogram is a rectangle, then its diagonals are congruent.
If the diagonals of a parallelogram are congruent, then it is a rectangle.
The diagonals of a rhombus are perpendicular.
If the diagonals of a parallelogram are perpendicular, then it is a rhombus.
Each diagonal of a rhombus bisects a pair of opposite angles.
Both pairs of base angles of an isosceles trapezoid are congruent.
The diagonals of an isosceles trapezoid are congruent.
The median of a trapezoid is parallel to the bases, and its measure is half of the sum of the bases.
Right Triangles & Trigonometry
82. Thm: (Pythagorean Theorem) – In a right triangle, a 2  b 2  c 2 , where c is the hypotenuse.
83. Thm: If the side measures of a triangle satisfy the Pythagorean Equation, then it is a right triangle
84. Thm: In a 45°-45°-90° isosceles right triangle, the hypotenuse is 2 times as long as a leg. (1,1, 2 )
85. 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. (1, 3 , 2)
Circles
86. Post: (Arc Addition) – Point Q is a point on PR
iff
mPQ  mQR  PR .
87. Thm: Two minor arcs are congruent iff their corresponding chords are congruent.
88. Thm: If a diameter is perpendicular to a chord, then it bisects the chord and its arc.
89. Thm: Two chords are congruent iff they are equidistant from the center.
1
90. Thm: If an angle is inscribed in a circle, then the angle measure =
the measure of the intercepted arc.
2
91. Thm: If two inscribed angles intercept the same or congruent arcs, then the angles are congruent.
92. Thm: If an inscribed angle intercepts a semicircle, then the angle is a right angle.
93. Thm: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
94. Thm: A line is tangent to a circle iff it is perpendicular to the radius at the point of tangency.
95. Thm: Two tangent segments drawn from the same exterior point are congruent.
1
96. Thm: If a secant and a tangent intersect at the point of tangency, then the angle measure = the
2
measure of the intercepted arc.
1
97. Thm: If two secants intersect in the interior of a circle, then the angle measure = the sum of the
2
measures of the arcs intercepted by the angle and its vertical partner.
98. Thm: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the
1
angle measure = the positive difference of the measures of the intercepted arcs.
2
99. Thm: If two chords intersect, then the products of the segments measures from each chord are equal.
100. Thm: If two secant segments are drawn to a circle from the same exterior point, then the product of
the measures of one full secant segment and its external segment is equal to the product of the measure
of the other full secant segment and its external segment.
101. Thm: If a tangent and a secant segment are drawn to a circle from the same exterior point, then the
square of the measure of the tangent segment is equal to the product of the measures of the full secant
segment and its external segment.
Polygons & Area
102. 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 = (n – 2)180.
103. Thm: (Exterior Angle Sum) – If a polygon is convex, then the sum of all exterior angle measures, one
at each vertex, is 360°.
104. Post: The area of a region is the sum of the areas of all of its non-overlapping parts.
105. Post: Congruent figures have equal areas.
Surface Area & Volume
106. 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.
Coordinate Geometry
107. 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.
108. Thm: Given two points A(x1, y1) and B(x2, y2) in a plane, the distance between A and B is given by the
equation: AB  ( x2  x1 )2  ( y2  y1 )2
109. Thm: Given two points A(x1, y1) and B(x2, y2) in a plane, the midpoint of segment AB is given by the
 x  x y  y2 
equation: mid   1 2 , 1

2 
 2