Warm Up A group of 7 people get

Warm Up A group of 7 people get together for a meeting. Before the meeting starts, they each shake hands with every person at the meeting. If a person only shakes hands with another person once, how many handshakes were there before the meeting began? 21

Lesson 2 -5 Review the properties of equality and use them to write proofs Identify the properties of equality and congruence

B + 6 B = = 11 5 How many squares are there in the bag?

Properties of Equality Addition Property If “a = b”, then “a + c = b + c” x– 6=2 Given Equation x – 6 + 6 =2 + 6 Addition Property EXAMPLE 1 x = 8 Subtraction Property EXAMPLE 2 Simplification If “a = b”, then “a – c = b – c” y+4=5 y + 4 – 4 =5 – 4 y = 1 Given Equation Subtraction Property Simplification

Multiplication Property EXAMPLE 3 w=3 2 2●w=3● 2 2 w = 6 Division Property EXAMPLE 4 If “a = b”, then “a ● c = b ● c” Given Equation Multiplication Property Simplification If “a = b”, then “a ÷ c = b ÷ c” 5 m = 10 Given Equation 5 m = 10 5 5 Division Property m = 2 Simplification

Let’s try some formal proofs, which are written in two columns Given: 3 x – 19 = 5, Prove: x = 8 Statements Reasons 1) 3 x – 19 = 5 1) Given 2) 3 x – 19 + 19 = 5 + 19 2) Addition Property 3) 3 x = 24 3) Simplification 4) 3 x = 24 3 3 4) Division Property 5) x = 8 5) Simplification

Solve 4 m – 8 = 2 m - 12, justify each step Statements Reasons 1) 4 m – 8 = 2 m - 12 1) Given 2) 4 m – 8 + 8 = 2 m – 12 + 8 2) Addition Property 3) 4 m = 2 m – 4 3) Simplification 4) 4 m – 2 m = 2 m – 4 – 2 m 4) Subtraction property 5) 2 m = -4 5) Simplification 6) 2 m = -4 2 2 6) Division property 7) m = -2 7) Simplification

Solve 2 p – 30 = -4 p + 6, justify each step Statements Reasons 1) 2 p – 30 = -4 p + 6 1) Given 2) 2 p – 30 + 30 = -4 p + 6 + 30 2) Addition Property 3) 2 p = -4 p + 36 3) Simplification 4) 2 p + 4 p = -4 p + 36 + 4 p 4) Addition property 5) 6 p = 36 5) Simplification 6) 6 p = 36 6 6 6) Division Property 7) 7) Simplification p=6

Reflexive Property EXAMPLE 5 AB 5 = = = “a = a” AB 5 Symmetric Property EXAMPLE If 6 2 =x x =2 If “a = b”, then “b = a” -5 = y y = -5 Transitive Property If “a = b”, and “b = c”, then “a = c” x=3 3=y and then x=y

Substitution Property EXAMPLE 7 If “a = b”, then “b substitutes a” x + y = 10 and y = 2 x + 2 = 10 Substitution Property x + 2 – 2 = 10 – 2 x = 8 Distributive Property Given Equation Subtraction Property Simplification If “a(b + c)”, then “a●b + a●c” Segment addition postulate If “point B lies on the line segment between points A and c”, then “AB + BC = AC” A B C

Solve 1 = x + 2 , justify each step 5 Statements Reasons 1) 1 = x + 2 5 1) Given 2) 5● 1 = 5●( x + 2) 5 2) Multiplication property 3) 1 = 5 x + 10 3) Distributive property 4) 1 – 10 = 5 x + 10 – 10 4) Subtraction Property 5) – 9 = 5 x 5) Simplification 6) 5 x = – 9 6) Symmetric property 7) 5 x = – 9 5 5 7) Division property 8) x=– 9 5 8) Simplification

Solve 9 C + 32 = F and C = 15 5 Statements 1) 9 C + 32 = F 5 2) C = 15 3 , justify each step Reasons 1) Given 2) Given 3) 9 (15) + 32 = F 5 3) Substitution property 4) 27 + 32 = F 4) Simplification 5) 59 = F 5) Simplification 6) F = 59 6) Symmetric property

Name the property that Justifies each statement. If CD = MN and CD = RS, then MN = RS A. Symmetric B. Substitution C. Reflexive

Name the property that Justifies each statement. If 2 = a, then a = 2 A. Symmetric B. Substitution C. Reflexive

Name the property that Justifies each statement. If x + y = 7, and y = 5 then x + 5 = 7 A. Transitive B. Substitution C. Reflexive

Name the property that Justifies each statement. If AB = CD, and CD = MN then AB = MN A. Transitive B. Substitution C. Symmetric

justify each step 3 x – 9 N Statements 2 x 4 x – 4 M P Reasons 1) NP = NM + MP 1) segment addition postulate 2) 4 x – 4 = 3 x – 9 + 2 x 2) Substitution property 3) 4 x – 4 = 5 x - 9 3) Simplification 4) 4 x – 4 x = 5 x – 9 – 4 x 4) Subtraction property 5) – 4= x-9 5) Simplification 6) – 4+9= x– 9+9 6) Addition property 7) 5= x 7) Simplification 8) x= 5 8) Symmetric property

Definition of congruent segments If “line segments are congruent”, then “they have the same length” A B C Angle addition postulate If “point D lies in the interior of angle ABC”, then “m ABD + m DBC = m ABC” A D 50 o 20 o B C

Given: H . justify each step X Statements 1) + = 2 xo 40 o Y Z Reasons 1) angle addition postulate 2) 2 x + 40 = 4 x + 20 2) Substitution property 3) 2 x + 40 – 20 = 4 x + 20 – 20 3) Subtraction Property 4) 2 x + 20 = 4 x 4) Simplification 5) 2 x + 20 – 2 x = 4 x – 2 x 5) Subtraction Property 6) 20 = 2 x 6) Simplification 7) 2 x = 20 7) Symmetric Property 8) x = 10 8) Division Property