Download Geometry Name Postulates, Theorems, Definitions, and Properties

Geometry
Name
Postulates, Theorems, Definitions, and Properties Used as
Reasons in Proofs (through chapter 2)
Postulates
1. Ruler Postulate – The points on a line can be matched one to one with real numbers. The real number that
corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is
the absolute value of the difference between the coordinates of A and B.
2. Segment Addition Postulate – If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is
between A and C.
ശሬሬሬሬԦ . The rays of the form ሬሬሬሬሬԦ
3. Protractor Postulate – Consider a point A on one side of ܱ‫ܤ‬
ܱ‫ ܣ‬can be matched one
to one with the real numbers from 0 to 180. The measure of ∠‫ ܤܱܣ‬is equal to the absolute value of the
ሬሬሬሬሬԦ .
difference between the real numbers for ሬሬሬሬሬԦ
ܱ‫ ܣ‬and ܱ‫ܤ‬
4. Angle Addition Postulate – If P is in the interior of ∠ܴܵܶ, then ݉∠ܴܵܲ + ݉∠ܲܵܶ = ݉∠ܴܵܶ.
5. Through any two points there exists exactly one line.
6. A line contains at least two points.
7. If two lines intersect, then their intersection is exactly one point.
8. Through any three noncollinear points there exists exactly one plane.
9. A plane contains at least three noncollinear points.
10. If two points lie in a plane, then the line containing them lies in the plane.
11. If two planes intersect, then their intersection is a line.
12. Linear Pair Postulate – If two angles form a linear pair, then they are supplementary.
Definitions
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Congruent Segments – Segments have equal length if and only if they are congruent.
Congruent Angles – Angles have equal measure if and only if they are congruent.
Midpoint – A point is a midpoint if and only if it divides a segment into two congruent segments.
Segment Bisector – A segment, ray, line, or plane is a segment bisector if and only if it divides a segment
into two congruent segments.
Angle Bisector – A ray is an angle bisector if and only if it divides an angle into two congruent angles.
Complementary angles – Two angles are complementary angles if and only if the sum of their measures is
90°.
Supplementary angles – Two angles are supplementary angles if and only if the sum of their measures is
180°.
Perpendicular lines – Two lines are perpendicular lines if and only if they intersect to form a right angle.
Right angle – An angle is a right angle if and only if it measures 90°.
Algebraic Properties of Equality
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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 – If a = b, then ac = bc
௔
௕
Division Property of Equality – If a = b and c ≠ 0, then ௖ = ௖
Substitution Property of Equality – If a = b, then a can be substituted for b in any equation or expression.
Distributive Property – ܽሺܾ + ܿሻ = ܾܽ + ܽܿ
Simplify – any time you add, subtract, multiply, or divide like terms on the SAME SIDE of the equation,
that is simply called “Simplify”
Reflexive Property of Equality
o For any real number a, a = a
o For any segment ‫ܤܣ‬, ‫ܤܣ = ܤܣ‬.
o For any angle A, ݉∠‫ܣ∠݉ = ܣ‬.
Symmetric Property of Equality
o If a = b, then b = a
o If ‫ܦܥ = ܤܣ‬, then ‫ܤܣ = ܦܥ‬.
o If ݉∠‫ܤ∠݉ = ܣ‬, then ݉∠‫ܣ∠݉ = ܤ‬.
Transitive Property of Equality
o If a = b and b = c, then a = c.
o If ‫ ܦܥ = ܤܣ‬and ‫ܨܧ = ܦܥ‬, then ‫ܨܧ = ܤܣ‬.
o If ݉∠‫ ܤ∠݉ = ܣ‬and ݉∠‫ܥ∠݉ = ܤ‬, then ݉∠‫ܥ∠݉ = ܣ‬.
Theorems
2.1 Properties of Segment Congruence – Segment congruence is reflexive, symmetric, and transitive.
തതതത ≅ തതതത
Reflexive: For any segment ‫ܤܣ‬, ‫ܤܣ‬
‫ܤܣ‬
തതതത
തതതത
Symmetric: If ‫ܦܥ ≅ ܤܣ‬, then ‫ ܤܣ ≅ ܦܥ‬.
Transitive: If തതതത
‫ ܦܥ ≅ ܤܣ‬and ‫ܨܧ ≅ ܦܥ‬, then തതതത
‫ܨܧ ≅ ܤܣ‬.
2.2 Properties of Angle Congruence – Angle congruence is reflexive, symmetric, and transitive.
Reflexive: For any angle ‫ܣ‬, ∠‫ܣ∠ ≅ ܣ‬
Symmetric: If ∠‫ܤ∠ ≅ ܣ‬, then ∠‫ܣ∠ ≅ ܤ‬
Transitive: If ∠‫ ∠ ≅ ܣ‬and ∠‫ܥ∠ ≅ ܤ‬, then ∠‫ܥ∠ ≅ ܣ‬
2.3 Right Angle Congruence Theorem – All right angles are congruent.
2.4 Congruent Supplements Theorem – If two angles are supplementary to the same angle (or to congruent
angles) then they are congruent.
2.5 Congruent Complements Theorem – If two angles are complementary to the same angle (or to congruent
angles) then the two angles are congruent.
2.6 Vertical Angles Theorem – Vertical angles are congruent.