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Algebraic Properties of Equality
For real numbers a, b, and c
Addition Property of Equality: if a=b , then a+c=b+c
Subtraction Property of Equality: if a=b , then a-c+=b-c
Multiplication Property of Equality: is a=b, then ac=bc
Division Property of Equality: if a=b and c≠0 , then (a/c)=(b/c)
Symmetric Property of Equality: if a=b, then b=a
Transitive Property of Equality: if a=b and b=c , then a=c
Reflexive Property of Equality: if a=a , then a=a
Substitution Property of Equality: if a=b, then a can be substituted for b
Distributive Property of Equality: a(b+c)=ab+ac
A figure is an angle if and only if it is composed of two rays which share a common endpoint. Each of these rays (or segments, as the case may be) is known as a side of the angle (For example, in the illustration at right), and the common point is known as the angle's vertex (point B in the illustration). Angles are measured by the difference of their slopes. The units for angle measure are radians and degrees. Angles may be classified by their degree measure.
- Acute Angle: an angle is an acute angle if and only if it has a measure of less than 90°
- Right Angle: an angle is an right angle if and only if it has a measure of exactly 90°
- Obtuse Angle: an angle is an obtuse angle if and only if it has a measure of greater than 90°
- Straight Angle: an angle is a straight angle if and only if it has a measure of exactly 180°
Using the Right Angle Congruence Theorem and the Straight Angle Congruence Theorem, all right angles and all straight angles are congruent
Angle Addition Postulate
If P is in the interior of an angle , then
The arithmetic mean of 2 numbers a and b can be calculated as: arithmetic mean=(a+b)/2
A figure bisects another figure if and only if it splits the figure it intersects into two equal parts
Center of a circle
Point P is the center of circle C if and only if all points in circle C are equidistant from point P and point P is contained in the same plane as circle C.
The set of all points in a plane that are equidistant from a given point (called the center of the circle).
The distance around a circle.
It is calculated as:
C=2πr (where r is the radius of the circle)
Two angles are complimentary if and only if the sum of their measures equals up to 90 degrees.
A polygon is said to be concave if and only if it contains at least one interior angle with a measure greater than 180° exclusively and less than 360° exclusively.
Two figures are congruent if and only if they have the same measure. It is designated by "≅".
Two angles formed by a transversal intersecting with two lines are corresponding angles if and only if one is on the inside of the two lines, the other is on the outside of the two lines, and both are on the same side of the transversal.
Corresponding Angles Postulate
If two lines cut by a transversal are parallel, then their corresponding angles are congruent.
Corresponding Parts of Congruent Triangles are Congruent Postulate
The Corresponding Parts of Congruent Triangles are Congruent Postulate (CPCTC) states:
- If ∆ABC ≅ ∆XYZ, then all parts of ∆ABC are congruent to their corresponding parts in ∆XYZ. For example:
- ∠ABC ≅ ∠XYZ
- ∠BCA ≅ ∠YZX
- ∠CAB ≅ ∠ZXY
CPCTC also applies to all other parts of the triangles, such as a triangle's altitude, median, circumcenter, et al.
A trigonometric ratio, abbreviated as cos.
A line segment is the diameter of a circle if and only if it is a chord of the circle which contains the circle's center.
- See Circle
Distance between 2 points can be calculated as the absolute value of the difference of the coordinates of the points.
In a coordinate plane, distance for points A(x1,y1) and B(x2,y2) can be calculated as:
The distance between a point and a line is measured by the perpendicular segments connecting the 2 (using the Perpendicular postulate).
For 2 positive numbers a and b, the geometric mean of a and b is the positive number x that satisfies (a/x)=(x/b). So, x2=√(ab)
geometric mean of a and b = (a/x)=(x/b) : x2=√(ab)
Height of a paralellogram
The perpendicular distance bewtween 2 bases of a paralellogram
In a right triangle, the side opposite of the right angle.
Using the Pythagorean Theorem, the measure of the hypotenuse can be calculated as:
c2=a2=b2 (where c is the hypotenuse and a and b are legs of the right triangle)
A collection of points is a line if and only if the collection of points is perfectly straight (aligned), is infinitely long, and is infinitely thin. Between any two points on a line, there exists an infinite number of points which are also contained by the line. Lines are usually written by two points in the line, such as line AB, or
A collection of points is a line segment if and only if it is perfectly straight, is infinitely thin, and has a finite length. A line segment is measured by the shortest distance between the two extreme points on the line segment, known as endpoints. Between any two points on a line segment, there exists an infinite number of points which are also contained by the line segment.
Interior angle of a regular polygon
The interior angles of a regular polygon are all congruent. As such, the measure of one interior angle of a regular polygon with n number of sides can be calculated as:
int angle = ((n-2)180)/n
A trapezoid in which the legs are congruent.
A triangle with at least 2 congruent sides.
Using the Base Angles theorem, the angles opposite of the congruent sides are also congruent.
The adjacent angles whose noncommon sides are opposite rays.
Using the Linear Pair Postulate, the angles in a linear pair are also supplementary.
If 2 angles forming a linear pair are congruent, then both the angles are right angles and the lines containing the angles are perpendicular.
An arc whose measure is greater than 180 degrees. It must be named by 3 points.
Midsegment of a Trapezoid
A segment that connects the midpoints of the legs of a trapezoid.
It is parallel to the bases and the length of it is the arithmetic mean of the measures of the bases.
An arc whose measure is less than 180 degrees.
Two lines or line segments are said to be parallel if and only if the lines are contained by the same plane and have no points in common if continued infinitely.
In a coordinate plane, two lines are parallel if and only if they share the same slope.
Two planes are said to be parallel if and only if the planes have no points in common when continued infinitely.
Two lines that intersect at a 90° angle.
In a coordinate plane, two lines are perpendicular if and only if their slopes' products are equal to -1 (or if the slopes are negative reciprocals).
Given a line, and a point P not in line , then there is one and only one line that goes through point P perpendicular to
An object is a plane if and only if it is a two-dimensional object which has no thickness or curvature and continues infinitely. A plane can be defined by three points. A plane may be considered to be analogous to a piece of paper.
A point is a zero-dimensional mathematical object representing a location in one or more dimensions. A point has no size; it has only location.
A polygon is a closed plane figure composed of at least 3 straight lines. Each side has to intersect another side at their respective endpoints, and that the lines intersecting are not collinear.
The radius of a circle is the distance between any given point on the circle and the circle's center.
All radii in the same circle (or congruent circles) have the same measure.
- See Circle
A ray is a straight collection of points which continues infinitely in one direction. The point at which the ray stops is known as the ray's endpoint. Between any two points on a ray, there exists an infinite number of points which are also contained by the ray.
A regular polygon is a polygon that is equilateral and equiangular.
The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the point's coordinate. The distance between two points is the absolute value of the difference between the two coordinates of the two points.
An arc whose measure is 180 degrees.
Sum of the interior angles of a polygon
The sum of the interior angles of a polygon with n number of sides is calculated as:
sum int angles = (n-2)180
Two angles are supplementary if and only if the sum of their measures equal up to 180 degrees.
- Geometry Main Page
- Geometry/Chapter 1 Definitions and Reasoning (Introduction)
- Geometry/Chapter 2 Proofs
- Geometry/Chapter 3 Logical Arguments
- Geometry/Chapter 4 Congruence and Similarity
- Geometry/Chapter 5 Triangle: Congruence and Similiarity
- Geometry/Chapter 6 Triangle: Inequality Theorem
- Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
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- Geometry/Chapter 9 Prisms, Pyramids, Spheres
- Geometry/Chapter 10 Polygons
- Geometry/Chapter 11
- Geometry/Chapter 12 Angles: Interior and Exterior
- Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
- Geometry/Chapter 14 Pythagorean Theorem: Proof
- Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
- Geometry/Chapter 16 Constructions
- Geometry/Chapter 17 Coordinate Geometry
- Geometry/Chapter 18 Trigonometry
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- Appendix C. Geometry/Postulates & Definitions
- Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry
- Part I- Euclidean Geometry:
- Chapter 1. Geometry/Points, Lines, Line Segments and Rays
- Chapter 2. Geometry/Angles
- Chapter 3. Geometry/Properties
- Chapter 4. Geometry/Inductive and Deductive Reasoning
- Chapter 5. Geometry/Proof
- Chapter 6. Geometry/Five Postulates of Euclidean Geometry
- Chapter 7. Geometry/Vertical Angles
- Chapter 8. Geometry/Parallel and Perpendicular Lines and Planes
- Chapter 9. Geometry/Congruency and Similarity
- Chapter 10. Geometry/Congruent Triangles
- Chapter 11. Geometry/Similar Triangles
- Chapter 12. Geometry/Quadrilaterals
- Chapter 13. Geometry/Parallelograms
- Chapter 14. Geometry/Trapezoids
- Chapter 15. Geometry/Circles/Radii, Chords and Diameters
- Chapter 16. Geometry/Circles/Arcs
- Chapter 17. Geometry/Circles/Tangents and Secants
- Chapter 18. Geometry/Circles/Sectors
- Appendix A. Geometry/Postulates & Definitions
- Appendix B. Geometry/The SMSG Postulates for Euclidean Geometry
- Part II- Coordinate Geometry:
- Two and Three-Dimensional Geometry and Other Geometric Figures
- Geometry/Perimeter and Arclength
- Geometry/Right Triangles and Pythagorean Theorem
- Geometry/2-Dimensional Functions
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- Geometry/Ellipsoids and Spheres
- Geometry/Coordinate Systems (currently incorrectly linked to Astronomy)
- Traditional Geometry:
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