Ch 3 Postulates Proofs and Equality Geometry St

Ch 3 Postulates, Proofs and Equality Geometry St. Barnabas H. S. Bronx, NY Geometry Lesson: First Postulates 1

Aim: What are the first postulates used in geometry proofs? Do Now: 1) You stand in front of a full-length mirror. How tall is your reflection? Ans. : Same height as myself 2) John is the same height as Lisa. What conclusion can you make? Ans. : Lisa is the same height as John. 3)Frank is the same age as Javier is the same age as Patricia. What conclusion can you make? Ans. : Patricia is the same age as Frank. 2

2 -Column Geometry Proofs Given: Information “given” to help start the proof. Prove: The final conclusion that we must make based on the given, postulates, definitions and theorems. Arrange proof in 2 -column table format. Every statement must have an accompanying reason. Statements 1)Given usually goes first. 2)Conclusions… 3)Conclusions… 4)Prove is last statement Reasons 1) Given… 2) Definitions, Postulates, Theorems… 3) Definitions, Postulates, Theorems… 4) Definitions, Postulates, Theorems… Geometry Lesson: First Postulates 3

Postulates and Theorems Defined Postulate: A postulate is a statement that we accept as true without proof. Theorem: A theorem is a statement that can be proved by deductive reasoning. Geometry Lesson: First Postulates 4

First Postulates -Properties of Equality Postulate #1 – Reflexive Property of Equality: “A quantity is equal to itself. ” a = a “A line segment (or angle) is congruent to itself. ” Ex 1: Ex 2: C B D A D B A Reflexive Postulate Geometry Lesson: First Postulates E C Reflexive Postulate 5

Postulate #2 – Symmetric Postulate of Equality: “An equality may be expressed in either order. ” If a = b, then, b = a Ex 1: Ex 2: Given: A B C D Symmetric Postulate P Q Symmetric Postulate Geometry Lesson: First Postulates 6

Postulate #3 – Transitive Postulate of Equality: If quantities are equal to the same quantity, then they are equal to each other. If a = b and b = c, then a = c C A Statements 1) 2) 3) 4) B Reasons Given Transitive Postulate Def. equilateral triangle Geometry Lesson: First Postulates 7

Example: Given: x Prove: Statements 1) 2) 3) 4) 5) y Reasons 1) 2) 3) 4) 5) Given Symmetric Postulate Transitive Postulate (Steps 1 & 3) Def. congruent angles Geometry Lesson: First Postulates 8

Examples: First Postulates 1) 2) A B C D L M R S C B A D E Geometry Lesson: First Postulates 9

Ex #1: A B C D L M R S Statements 1) 2) 3) 4) 5) 6) 7) Reasons 1) 2) 3) 4) 5) 6) 7) Given Transitive Postulate Given Symmetric Postulate Transitive Postulate Def. congruent line segments Geometry Lesson: First Postulates 10

C Ex#2 B A D E Statements 1) 2) 3) 4) 5) 6) Reasons 1) 2) 3) 4) 5) 6) Given Def. Angle Bisector Transitive Postulate (3, 5) Geometry Lesson: First Postulates 11