CK-12 Geometry-Concepts

2.8. Two-Column Proofs http://www.ck12.org

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∼=.

Example C

TheRight Angle Theoremstates that if two angles are right angles, then the angles are congruent. Prove this
theorem.

To prove this theorem, set up your own drawing and name some angles so that you have specific angles to talk about.

Given:^6 Aand^6 Bare right angles

Prove:^6 A∼=^6 B

TABLE2.21:

Statement Reason

1.^6 Aand^6 Bare right angles 1. Given
2.m^6 A= 90 ◦andm^6 B= 90 ◦ 2. Definition of right angles
3.m^6 A=m^6 B 3. TransitivePoE
4.^6 A∼=^6 B 4.∼=angles have = measures

Any time right angles are mentioned in a proof, you will need to use this theorem to say the angles are congruent.