2.4 Reasoning with Properties from Algebra ?

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…

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

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

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

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)

Ex: Solve the equation & write a reason for each step. 1. 2(3x+1) = 5x x+2 = 5x x+2 = x = Given 2. Distributive prop 3. Subtraction prop of = 4. Subtraction prop of =

Solve 55z-3(9z+12) = -64 & write a reason for each step z-3(9z+12) = z-27z-36 = z-36 = z = z = Given 2. Distributive prop 3. Simplify (or collect like terms) 4. Addition prop of = 5. Division prop of =

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

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

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

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° Identifying Property of Equality and Congruence Symm. Prop. of = Trans. Prop of  Reflex. Prop. of = Reflex. Prop. of .

Example from scratch…

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 Vertical angles

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

2.5 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° ° 40° 34 75° 105°

Identifying Angle Pairs In the diagram identify pairs of numbered angles that are related as follows: a. Complementary b. Supplementary c. Vertical d. Adjacent

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

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

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

Theorems About Angles Theorem 2-3Congruent 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

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 = 180. 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

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 __________.

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