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2.5 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 a b c c 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 (mirror) AB ≅ AB ∠A≅∠A Symmetric Property (twins) If AB ≅ CD, then CD ≅ AB If ∠ A ≅ ∠ B, then ∠ B ≅ ∠ A Transitive Property (triplets) If AB ≅ CD and CD ≅ EF, then AB ≅ EF If ∠ A ≅ ∠ B and ∠ B ≅ ∠ C, then ∠ A ≅ ∠ C Ex: Solve the equation & write a reason for each step. 1. 2(3x+1) = 5x+14 2. 6x+2 = 5x+14 1. Given 2. Distributive prop 3. x+2 = 14 4. x = 12 3. Subtraction prop of = 4. Subtraction prop of = Solve 55z-3(9z+12) = -64 & write a reason for each step. 1. 2. 3. 4. 5. 55z-3(9z+12) = -64 55z-27z-36 = -64 28z-36 = -64 28z = -28 z = -1 1. 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 4x – 4 = 5x – 9 Simplify. –4 = x – 9 5=x Segment Addition Post. Subtraction Property of Equality Addition Property of Equality Solve, Write a justification for each step. mABC = mABD + mDBC 8x° = (3x + 5)° + (6x – 16)° Add. Post. Subst. Prop. of Equality 8x = 9x – 11 Simplify. –x = –11 Subtr. Prop. of Equality. x = 11 Mult. Prop. of Equality. Remember! Numbers are equal (=) and figures are congruent (). Identifying Property of Equality and Congruence Identify the property that justifies each statement. A. QRS QRS Reflex. Prop. of . B. m1 = m2 so m2 = m1 Symm. Prop. of = C. AB CD and CD EF, so AB EF. Trans. Prop of D. 32° = 32° Reflex. Prop. of = Example from scratch… 2.6 Proving Angles Congruent 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 1 4 <2 and <4 are Vertical angles 2 3 Proving Angles Congruent • Adjacent Angles: Two coplanar angles that share a side and a vertex 1 2 <1 and <2 are Adjacent Angles 1 2 Proving Angles Congruent • Complementary Angles: Two angles whose measures have a sum of 90° 50° 2 40° 1 • Supplementary Angles: Two angles whose measures have a sum of 180° 105° 3 4 75° Identifying Angle Pairs In the diagram identify pairs of numbered angles that are related as follows: a. Complementary 1 b. Supplementary c. Vertical d. Adjacent 5 2 4 3 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 2-1 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 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 Proving Theorems Paragraph Proof: Written as sentences in a paragraph Given: <1 and <2 are vertical angles Prove: <1 = <2 1 3 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: Prove: <1 and <2 are supplementary <3 and <2 are supplementary <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 __________.