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Proving Theorems

In this lesson on Pre-AP Geometry, we explore geometric proofs using deductive reasoning. We start by defining key concepts like midpoint, segment bisector, and angle bisector, followed by their respective theorems. We'll utilize the Midpoint Theorem to understand how to find the midpoint of a segment given its endpoints, and we will apply the Angle Bisector Theorem to demonstrate the properties of angle bisectors. Deductive reasoning is emphasized to show how conclusions logically follow from premises. Practice exercises included.

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Proving Theorems

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  1. Proving Theorems Lesson 2.3 Pre-AP Geometry

  2. 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.

  3. 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.

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

  5. 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)

  6. 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:

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

  8. 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).

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

  10. 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 B C

  11. Proof: Angle Bisector Theorem Given: BXis the bisector of ∠ABC. Prove: m ∠ABX = ½ m ∠ABC; m ∠XBC = ½m ∠ABC

  12. 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 .

  13. 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.”

  14. 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

  15. Computer Key-In Project

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