Download Definitions, Postulates, Properties, and Theorems

Definitions, Postulates, Properties, and Theorems
DEFINITIONS:
Segments
Definition of congruent segments:
Definition of a midpoint:
If AB ≅ CD then, AB = CD
If M is the midpoint of AB , then AM = MB
Definition of a segment bisector: Any point, line, ray or plane that divides a segment into two congruent segments. If
line AB bisects segment CD at point E, then CE = ED.
Angles
If m∠A = m∠B, then∠A ≅ ∠B
Definition of congruent angles:
Definition of complementary angles: If ∠Aand ∠B are complementary, then m∠A + m∠B = 90o
Definition of supplementary angles: If ∠Aand ∠B are supplementary, then m∠A + m∠B = 180o
Definition of an angle bisector:
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If BCbi sec ts∠ABD , then m∠ABC = m∠CBD
Definition of a Right Angle:
If ∠A is a right angle, the m∠A = 90o
Definition of Perpendicular Lines
HJJG HJJG
HJJG
HJJG
If AB ⊥ CD, then ABandCD intersect to form a right angle.
Definition of Linear Pair:
A pair of adjacent angles with non-common sides on same line
Distance from point to a line: the length of the perpendicular segment from the point to the line.
POSTULATES
Segment Addition Postulate: Two small segments equal a big segment
Angle Addition Postulate:
Two small angles make a big angle
Linear Pair Postulate:
If two angles form a linear pair, then they are supplementary
PROPERTIES OF EQUALITY (POE)
Addition Property:
If a = b, then a + c = b + c
Subtraction Property:
If a = b, then a – c = b – c
Multiplication Property:
If a = b, then ac = bc
a b
=
c c
Division Property:
If a = b, and c is NOT equal to 0, then
Substitution Property
If a = b, then a can be substituted for b in any equation or expression
Distributive Property
a(b + c) = ab + ac, where a, b, and c are real numbers.
Reflexive Property of Equality
Real Numbers: For any real number a, a = a
Segment Length: For any segment AB, AB = AB
Angle Measure: For any angle A, A = A
Symmetric Property of Equality
Real Numbers: For any real numbers a and b, If a = b, then b = a
Segment Lengths: For any segment AB and CD, If AB = CD, then CD = AB
Angle Measure: For any angles A and B, If m∠A = m∠B , then m∠B = m∠A
Transitive Property of Equality
Real numbers: For any real numbers a, b, and c, if a = b, and b = c, then a = c
Segment Lengths: For any segments AB, CD, and EF, if AB = CD and CD = EF, then AB = EF
Angle Measure: For any angles A, B, and C, if m∠A = m∠B and m∠B = m∠C , then m∠A = m∠C
PROPERTIES OF CONGRUENCE (POC)
Reflexive Property of Congruence:
Segments: For any segment AB, AB ≅ AB
Angles: For any angle A, ∠A = ∠A
Symmetric Property of Congruence:
Segments: If AB ≅ CD , then CD ≅ AB
Angles: If ∠A ≅ ∠B , then ∠B ≅ ∠A
Transitive Property of Congruence:
Segment Lengths: For any segments AB, CD, and EF, if AB ≅ CD and CD ≅ EF, then AB ≅ EF
Angle Measure: For any angles A, B, and C, if m∠A ≅ m∠B and m∠B ≅ m∠C , then m∠A ≅ m∠C
ANGLE THEOREMS
Right Angle Congruence Theorem: All right angles are congruent
Congruent Supplements Theorem: If m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180 , then ∠1 ≅ ∠3 .
Congruent Complements Theorem: If m∠4 + m∠5 = 90 and m∠5 + m∠6 = 90 , then ∠4 ≅ ∠6 .
Vertical Angles Theorem: Vertical angles are congruent.
Special Angle Relationships
Definition of a Transversal: A line that intersects two or more coplanar lines at different points, forming 8
angles. Special angle pairs are created.
•
Corresponding Angles occupy corresponding positions
•
Alternate Exterior Angles lie outside the two lines on opposite sides of the transversal.
•
Alternate Interior Angles lie between the two lines on opposite sides of the transversal.
•
Consecutive Interior Angles lie between the two lines on the same side of the transversal.
If the 2 lines are parallel, then Corresponding Angles, Alternate Exterior, and Alternate Interior Angles are
congruent, and Consecutive Interior Angles are supplementary.