Download Division Property of Equality.

2.4 Reasoning in Algebra
Properties
Utilize the following properties throughout your studies in this section.
The last three properties may seem new, but are very useful and handy.
Example #1
Solve for X by filling in the property that
justifies each step.
So, starting off with the first step. We can justify a. without
looking at the numbers and variables given. You simply add up
the angles to get 180 degrees, a straight line. The justification
would be Angle Addition Postulate …..
Therefore, a. Angle Addition Postulate.
Second Step (b): The justification has to be a property where
something replaces another, otherwise known as …..
Subsitution property. Clearly in the diagram, angle CDE is also x
and angle EDF is (3x+20).
Third Step (c): Try to solve this step…………………………
You may think that this is addition property, but you’re wrong.
Because the variables are the same, the parentheses are
unneeded so this is simply a case of Simplification.
Fourth Step (d): In this step, you need to reduce 4x+20=180 to
4x=160. This means you need to subtract 20 from both sides.
This calls for the Subtraction Property of Equality.
Fifth Step (e): Last Step! You need to produce x= 40 from
4x=160. With basic algebra, you can realize this takes division.
Hence, the Division Property of Equality.
Now a bit harder and longer,
Example #2
A bit more difficult, Example #3.
a. Segment Addition Postulate*
b. Substitution Property of Equality
c. Distributive Property
d. Simplify
e. Subtraction Property of Equality
f. Division Property of Equality
*If three points, A, B, and C, are collinear, and B is
between points A and C, then AB+BC=AC.
Last Example! #3
Which property justifies this statement? In this situation,
the BC is being added to the right side and subtracted
from the left. This is known as Addition Property of
congruence.
Now, do it on your own with these 3 practice problems!
1)
2)
3)
Answer Key
1) D
2) A. mEPF+mFPG=mEPG by Angle Addition Postulate B.
mEPF=mFPG by Definition of Angle Bisector C. 4x+2=6x-10
by Substitution Property of Equality D. 4x+12=6x by
Addition Property of Equality E. 12=2x by Subtraction
Property of Equality F. 6=x by Division Property of Equality
3) In the fifth step, the Distributive Property should have
simplified to ab-a2=b2-a2.
2.5 Proving Angles Congruent
Definitions
Theorem: A conjecture (a conclusion reached by using inductive
reasoning) that is proven
Paragraph proof: a proof written as sentences in a paragraph
Theorems
Theorem 2.1 Vertical Angles Theorem
Vertical angles are congruent.
Theorem 2.2 Congruent Supplements Theorem
If two angles are supplements of the same angle (or of
congruent angles), then the two angles are congruent.
Theorem 2.3 Congruent Complements Theorem
If two angles are complements of the same angle (or of
congruent angles), then the two angles are congruent.
Theorem 2.4
All right angles are congruent.
Theorem 2.5
If two angles are congruent and supplementary, then each is a
right angle.
Remember, in a proof, information in the “Given” supplies what
you know. The “Prove” section supplies what you need to
show, ultimately completing your proof.
Example #1 using Vertical Angles Theorem
Solve for x.
Example #2
a. 90
b. 90
X+10=4x-35
Vertical Angles are congruent
10=3x-35
Subtraction Property of Equality
45=3x
Addition Property of Equality
15=x
Division Property of Equality
c. Substitution Property of Equality
d. m3
Example #3 using Theorem 2.5
a. V
b. 180 (Definition of supplementary angles)
c. Division (NOT MULTIPLICATION!)
d. right
Now time for practice!
1) Find the value of x.
2) Write a paragraph proof with the following information.
3)Find the value of each variable and the measure of each
labeled angle.
Answer Key
1) X=15
2) Because of Vertical Angles Theorem, angles 2 and 3 are
congruent. Since the Given states that angle 1 is
complementary to angle 2, you can say that angle 1 is
complementary to angle 3 by Substitution Property. Also since
angle 3 is complementary to angle 4, and angle 1 and 3 are
complements, then bby Congruent Complements Theorem, you
can state that angles 1 and 4 are congruent. Angle 5 is
supplementary to angle 1 by Definition of Supplementary
Angles and so are angles 4 and 6. Interchanging angles 1 and 4
by Substitution property, angles 5 and 6 are both
supplementary to angle 4. Angles 5 and 6 are congruent by
Congruent Supplements Theorem.
3)Y = 70, X=35, 2X= 70, X+Y+5=110