Geometry: Postulates, Theorems, and Proofs (Page 1 Notes)
Midpoint
- Definition: M is the midpoint of \overline{AB} ⇔ AM = MB
Midpoint Theorem
- M is the midpoint of \overline{AB} ⇔ AM \cong MB
Reflexive
Segment Addition Postulate
Supplementary Angles
- m\angle A + m\angle B = 180^\circ
Supplement Theorem
- If two angles form a linear pair, then they are supplementary.
Symmetric
- If a = b, then b = a
- If a \cong b, then b \cong a
Angle Addition Postulate
- m\angle ABC + m\angle CBD = m\angle ABD
Complementary Angles
- m\angle A + m\angle B = 90^\circ
Complement Theorem
- If two angles form a right angle, then they are complementary.
Transitive
- If a = b and b = c, then a = c
- If a \cong b and b \cong c, then a \cong c
Congruent Segments
Congruent Supplements
- If two angles are supplementary to the same angle, then they are congruent.
Addition
- If a = b, then a + c = b + c
Congruent Angles
- \angle A \cong \angle B \iff m\angle A = m\angle B
Congruent Complements
- If two angles are complementary to the same angle, then they are congruent.
Subtraction
- If a = b, then a - c = b - c
Vertical Angles
- Vertical angles: two nonadjacent angles formed by intersecting lines.
- Vertical Angles Theorem: vertical angles are congruent.
Multiplication
Right Angle
- A right angle: an angle with measure 90^\circ.
- All right angles are congruent.
Division
- If a = b, then \dfrac{a}{c} = \dfrac{b}{c}
Linear Pair
- A pair of adjacent angles whose non-common sides are opposite rays.
Perpendicular
- Perpendicular lines form 4 right angles.
Substitution
- If a = b, then a may replace b.
Segment Bisector
- A segment, line, or plane that intersects a segment at its midpoint.
CPCTC
- Corresponding Parts of Congruent Triangles are Congruent.
Distributive
Angle Bisector
- A ray that divides an angle into two congruent angles.
Reasons for Proofs
- Justifications used in geometric proofs (definitions, postulates, theorems, and previously proven statements).
