CK-12 Geometry Concepts

http://www.ck12.org Chapter 2. Reasoning and Proof

Example A

Write a two-column proof for the following:

IfA,B,C, andDare points on a line, in the given order, andAB=CD, thenAC=BD.

First of all, when the statement is given in this way, the “if” part is the given and the “then” part is what we are trying
to prove.

Always start with drawing a picture of what you are given.

Plot the points in the orderA,B,C,Don a line.

Add the corresponding markings,AB=CD, to the line.

Draw the two-column proof and start with the given information.From there, we can use deductive reasoning
to reach the next statement and what we want to prove.Reasons will be definitions, postulates, properties and
previously proven theorems.

TABLE2.19:

Statement Reason

1.A,B,C, andDare collinear, in that order. 1. Given

2.AB=CD 2. Given

3.BC=BC 3. Reflexive PoE

4.AB+BC=BC+CD 4. Addition PoE

5.AB+BC=AC,BC+CD=BD 5. Segment Addition Postulate

6.AC=BD 6. Substitution or Transitive PoE

When you reach what it is that you wanted to prove, you are done.

Example B

Write a two-column proof.

Given:

−→

BFbisects^6 ABC;^6 ABD∼=^6 CBE

Prove:^6 DBF∼=^6 EBF

First, put the appropriate markings on the picture. Recall, that bisect means “to cut in half.” Therefore, if

−→

BFbisects

(^6) ABC, thenm (^6) ABF=m (^6) F BC. Also, because the word “bisect” was used in the given, the definition will probably
be used in the proof.
TABLE2.20:
Statement Reason

−→

BFbisects^6 ABC,^6 ABD∼=^6 CBE 1. Given

2.m^6 ABF=m^6 F BC 2. Definition of an Angle Bisector

3.m^6 ABD=m^6 CBE 3. If angles are∼=, then their measures are equal.

4.m^6 ABF=m^6 ABD+m^6 DBF,m^6 F BC=m^6 EBF+

m^6 CBE

4. Angle Addition Postulate

5.m^6 ABD+m^6 DBF=m^6 EBF+m^6 CBE 5. Substitution PoE

6.m^6 ABD+m^6 DBF=m^6 EBF+m^6 ABD 6. Substitution PoE

7.m^6 DBF=m^6 EBF 7. Subtraction PoE

8.^6 DBF∼=^6 EBF 8. If measures are equal, the angles are∼=.