1. Lesson 2.3 Pre-AP Geometry
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 C B

11. 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

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:

15. Computer Key-In Project