1. 2.5 Reasoning with Properties from Algebra?
Reasoning with Properties from Algebra

2. What are we doing, & Why are we doing this?
In algebra, you did things because you were told to….
In geometry, we can only do what we can PROVE…
We will start by justifying algebra steps (because we already know how)
Then we will continue justifying steps into geometry…

3. But first…we need to
1. Learn the different properties / justifications
2. Know format for proving / justifying mathematical statements
3. Apply geometry properties to proofs

4. Properties of Equality (from algebra)
Addition property of equality- if a=b, then a+c=b+c.
(can add the same #, c, to both sides of an equation)
Subtraction property of equality - If a=b, then a-c=b-c.
(can subtract the same #, c, from both sides of an equation)
Multiplication prop. of equality- if a=b, then ac=bc.
Division prop. of equality- if a=b, then

5. Properties of Equality (Algebra)
Reflexive prop. of equality- a=a
Symmetric prop of equality- if a=b, then b=a.
Transitive prop of equality- if a=b and b=c, then a=c.
Substitution prop of equality- if a=b, then a can be plugged in for b and vice versa.
Distributive prop.- a(b+c)=ab+ac OR (b+c)a=ba+ca

6. Properties of Equality (geometry)
Reflexive Property
AB ≅ AB
∠ A ≅ ∠ A
Symmetric Property
If AB ≅ CD, then CD ≅ AB
If ∠ A ≅ ∠ B, then ∠ B ≅ ∠ A
Transitive Property
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
If ∠ A ≅ ∠ B and ∠ B ≅ ∠ C, then ∠ A ≅ ∠ C
(mirror)
(twins)
(triplets)

7. Ex: Solve the equation & write a reason for each step.
2(3x+1) = 5x+14
6x+2 = 5x+14
x+2 = 14
x = 12
Given
Distributive prop
Subtraction prop of =

8. Solve 55z-3(9z+12) = -64 & write a reason for each step.
Given
Distributive prop
Simplify (or collect like terms)
Addition prop of =
Division prop of =

9. Solving an Equation in Geometry with Justifications
NO = NM + MO
Segment Addition Post.
4x – 4 = 2x + (3x – 9)
Substitution Property of Equality
4x – 4 = 5x – 9
Simplify.
–4 = x – 9
Subtraction Property of Equality
5 = x
Addition Property of Equality 9

10. Solve, Write a justification for each step.  Add. Post.
mABC = mABD + mDBC
8x° = (3x + 5)° + (6x – 16)°
Subst. Prop. of Equality
8x = 9x – 11
Simplify.
–x = –11
Subtr. Prop. of Equality.
x = 11
Mult. Prop. of Equality. 10

11. Numbers are equal (=) and figures are congruent ().
Remember! 11

12. Identifying Property of Equality and Congruence
Identify the property that justifies each statement.
A. QRS  QRS
B. m1 = m2 so m2 = m1
C. AB  CD and CD  EF, so AB  EF.
D. 32° = 32°
Reflex. Prop. of .
Symm. Prop. of =
Trans. Prop of 
Reflex. Prop. of = 12

13. Example
from
scratch…

14. 2.6 Proving Angles Congruent

15. Proving Angles Congruent
Vertical Angles: Two angles whose sides form two pairs of opposite rays; form two pairs of congruent angles
<1 and <3 are
Vertical angles
<2 and <4 are 1 4 2 3

16. Proving Angles Congruent
Adjacent Angles: Two coplanar angles that share a side and a vertex 1 2 1
<1 and <2 are
Adjacent Angles 2

17. Proving Angles Congruent
Complementary Angles: Two angles whose measures have a sum of 90°
Supplementary Angles: Two angles whose measures have a sum of 180°
50° 2
40° 1
105°
75° 3 4

18. Identifying Angle Pairs
In the diagram identify pairs of numbered angles that are related as follows:
Complementary
Supplementary
Vertical
Adjacent 2 1 3 5 4

19. Making Conclusions
Whether you draw a diagram or use a given diagram, you can make some conclusions directly from the diagrams. You CAN conclude that angles are
Adjacent angles
Adjacent supplementary angles
Vertical angles

20. Making Conclusions
Unless there are markings that give this information, you CANNOT assume
Angles or segments are congruent
An angle is a right angle
Lines are parallel or perpendicular

21. Theorems About Angles
Theorem 2-1 Vertical Angles Theorem Vertical Angles are Congruent Theorem 2-2 Congruent Supplements If two angles are supplements of the same angle or congruent angles, then the two angles are congruent

22. Theorems About Angles
Theorem 2-3 Congruent Complements If two angles are complements of the same angle or 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, each is a right angle

23. Proving Theorems Paragraph Proof: Written as sentences in a paragraph
Given: <1 and <2 are
vertical angles
Prove: <1 = <2
Paragraph Proof: By the Angle Addition Postulate, m<1 + m<3 = 180 and m<2 + m<3 = By substitution, m<1 + m<3 = m<2 + m<3. Subtract m<3 from each side. You get m<1 = m<2, which is what you are trying to prove. 1 3 2

24. Proving Theorems
Given: <1 and <2 are supplementary <3 and <2 are supplementary Prove: <1 = <3 Proof: By the definition of supplementary angles, m<___ + m<____ = _____ and m<___ + m<___ = ____. By substitution, m<___ + m<___ = m<___ + m<___. Subtract m<2 from each side. You get __________.