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Geometry Postulates and Theorems Explained

Learn about postulates that define geometric relationships, theorems that prove true statements, and properties like Segment Addition and Angle Addition. Practice solving equations and identifying congruence properties. Discover essential angle theorems.

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Geometry Postulates and Theorems Explained

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  1. Properties of Equality and Proving Segment &Angle Relationships Section 2-6, 2-7, 2-8

  2. Postulate - A statement that describes the relationship between basic terms in Geometry. Postulates are accepted as true without proof. Examples of some Postulates: • Through any 2 points there is exactly 1 line. • Through any 3 noncollinear points there is exactly 1 plane. • A line contains at least 2 points. • A plane contains at least 3 noncollinear points.

  3. Theorem A conjecture or statement that can be shown to be true. Used like a definition or postulate. Midpoint Theorem - If M is the midpoint of AB, then AM to MB. A M B

  4. Proof A logical argument in which each statement is supported by a statement that is true (or accepted as true). Supporting evidence in a proof (the reason you can make the statement) are usually postulates, theorems, properties, definitions or given information.

  5. Remember! The Distributive Property states that a(b + c) = ab + ac. Properties of Equality

  6. Segment Addition Postulate • If B is between A and C, then AB + BC = AC Converse: If AB + BC = AC, then B is between A and C. A B C

  7. Example: Solving an Equation in Geometry Write a justification for each step.

  8. Example: Solving an Algebraic Equation Write a justification for each step. 3(x - 2) = 42

  9. Statement Reason Example: Proving an Algebraic Conditional StatementWrite a justification for each step.If 3(x - 5/3) = 1, then x=2

  10. Angle Addition Postulate • If R is in the interior of PQS, then mPQR + mRQS = m PQS. • Converse: If mPQR + mRQS = m PQS, then R is in the interior of PQS P R Q S

  11. Segment and Angle Congruence Theorems - Congruence of Segments (or Angles) is Reflexive, Symmetric and Transitive

  12. Example: Identifying Property of Equality and Congruence Identify the property that justifies each statement. A. QRS  QRS B. m1 = m2 so m2 = m1 C. AB  CD and CD  EF, so AB EF. D. 32° = 32° Reflex. Prop. of . Symm. Prop. of = Trans. Prop of  Reflex. Prop. of =

  13. Practice Complete each sentence. 1.If the measures of two angles are ? , then the angles are congruent. 2. If two angles form a ? , then they are supplementary. 3. If two angles are complementary to the same angle, then the two angles are ? . equal linear pair congruent

  14. Additional Angle Theorems that You Should Know • Supplement Theorem – If two angles form a linear pair, then they are supplementary angles. • Complement Theorem – If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.

  15. Three More Angle Theorems that You Should Know • Angles supplementary to the same angle, or to congruent angles, are congruent. • Angles complementary to the same angle, or to congruent angles, are congruent. • Vertical Angles Theorem – If two angles are vertical angles, then they are congruent.

  16. Right Angles Theorems that Are Important: • Perpendicular lines intersect to form four right angles. • All right angles are congruent. • Perpendicular lines form congruent adjacent angles. • If two angles are congruent and supplementary, then each is a right angle. • If two congruent angles form a linear pair, then they are right angles.

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