CK-12 Geometry - Second Edition

2.4. Algebraic and Congruence Properties http://www.ck12.org

2 ( 3 x− 4 )+ 11 =x− 27

6 x− 8 + 11 =x− 27 Distributive Property

6 x+ 3 =x− 27 Combine like terms

6 x+ 3 − 3 =x− 27 − 3 Subtraction PoE

6 x=x− 30 Simplify

6 x−x=x−x− 30 Subtraction PoE

5 x=− 30 Simplify

5 x

5

=

− 30

5

Division PoE

x=− 6 Simplify

Example 2:Given pointsA,B, andC, withAB= 8 ,BC=17, andAC=20. AreA,B, andCcollinear?

Solution:Set up an equation using the Segment Addition Postulate.

AB+BC=AC Segment Addition Postulate

8 + 17 = 20 Substitution PoE

256 = 20 Combine like terms

Because the two sides are not equal,A,BandCare not collinear.

Example 3:Ifm^6 A+m^6 B= 100 ◦andm^6 B= 40 ◦, prove that^6 Ais an acute angle.

Solution:We will use a 2-column format, with statements in one column and their corresponding reasons in the
next. This is formally called a 2-column proof.

TABLE2.15:

Statement Reason

1.m^6 A+m^6 B= 100 ◦andm^6 B= 40 ◦ Given(always the reason for using facts that are told to

us in the problem)

2.m^6 A+ 40 ◦= 100 ◦ Substitution PoE

3.m^6 A= 60 ◦ Subtraction PoE

4.^6 Ais an acute angle Definition of an acute angle,m^6 A< 90 ◦

Two-Column Proof

Example 4:Write a two-column proof for the following: