Properties of Equality and Proving Segment & Angle Relationships

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.