Lesson 2.3 Pre-AP Geometry Proving Theorems Lesson 2.3 Pre-AP Geometry

Proofs Geometric proof is deductive reasoning at work. Throughout a deductive proof, the “statements” that are made are specific examples of more general situations, as is explained in the "reasons" column. Recall, a theorem is a statement that can be proved.

Vocabulary Midpoint The point that divides, or bisects, a segment into two congruent segments. Bisect To divide into two congruent parts. Segment Bisector A segment, line, or plane that intersects a segment at its midpoint.

Midpoint Theorem If M is the midpoint of AB, then AM = ½AB and MB = ½AB

Proof: Midpoint Formula Given: M is the midpoint of Segment AB Prove: AM = ½AB; MB = ½AB Statement 1. M is the midpoints of segment AB 2. Segment AM= Segment MB, or AM = MB 3. AM + MB = AB 4. AM + AM = AB, or 2AM = AB 5. AM = ½AB  6. MB = ½AB Reason 1. Given 2. Definition of midpoint 3. Segment Addition Postulate 4. Substitution Property (Steps 2 and 3) 5. Division Prop. of  Equality 6. Substitution Property. (Steps 2 and 5)

The Midpoint Formula The Midpoint Formula If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint of segment AB has coordinates:

The Midpoint Formula Application: Find the midpoint of the segment defined by the points A(5, 4) and B(-3, 2).

Midpoint Formula Application: Find the coordinates of the other endpoint B(x, y) of a segment with endpoint C(3, 0) and midpoint M(3, 4).

Vocabulary Angle Bisector A ray that divides an angle into two adjacent angles that are congruent.

Angle Bisector Theorem If BX is the bisector of ∠ABC, then the measure of ∠ABX is one half the measure of ∠ABC and the measure of ∠XBC one half of the ∠ABC. A X C B

Proof: Angle Bisector Theorem Given: BX is the bisector of ∠ABC. Prove: m ∠ABX = ½ m ∠ABC; m ∠XBC = ½m ∠ABC Statement Reason 1. BX is the bisector of ∠ABC 1. Given 2. m ∠ABX + m ∠XBC = m ∠ABC 2. Angle addition postulate 3. m∠ ABX = m ∠XBC 3. Definition of bisector of an angle 4. m∠ ABX + m ∠ABX = 2 m ∠ABC; m ∠XBC = m ∠XBC =2 m ∠ABC 4. Addition property 5. m ∠ABX = ½ m ∠ABC; m ∠XBC = ½ m ∠ABC 5. Division property

Deductive Reasoning If we take a set of facts that are known or assumed to be true, deductive reasoning is a powerful way of extending that set of facts. In deductive reasoning, we say (argue) that if certain premises are known or assumed, a conclusion necessarily follows from these. Of course, deductive reasoning is not infallible: the premises may not be true, or the line of reasoning itself may be wrong .

Deductive Reasoning For example, if we are given the following premises: A) All men are mortal, B) and Socrates is a man, then the conclusion Socrates is mortal follows from deductive reasoning. In this case, the deductive step is based on the logical principle that "if A implies B, and A is true, then B is true.”

Written Exercises Problem Set 2.3A, p. 46: # 1 – 12 Problem Set 2.3B, P. 47: # 13 – 22 Challenge: p.48, Computer Key-In Project (optional) Submit a print out of your results from running the program along with your answers to Exercises 1 – 3. Download BASIC at: http://www.justbasic.com

Computer Key-In Project