Download Class Notes 2.5 - Geometry PreAP GT

Class Notes 2.5
2.5: Reason Using Properties from Algebra
When you solve an equation, you use properties of real numbers. Since segment lengths and angle measures are also real numbers, you can use these properties to write logical arguments about geometric figures.
Algebraic Properties of Equality
Let a, b, and c be real numbers.
Addition Property
If a = b, then a + c = b + c
Subtraction Property
If a = b, then a ­ c = b ­ c
Multiplication Property
If a = b, then ac = bc
Division Property
If a = b and c ≠ 0, then a = b
Substitution Property
If a = b, then a can be substituted for b in any equation or expression.
c
c
1
Class Notes 2.5
1. Solve 3x + 8 = ­4x ­ 34. Write the reasons for each step.
Equation
Reason
3x + 8 = ­4x ­ 34
Given
Distributive Property
a(b + c) = ab + ac, where a, b, and c are real numbers
2. Solve 60 = ­3(8x ­ 4). Write a reason for each step.
Equation
Reason
60 = ­3(8x ­ 4)
Given
2
Class Notes 2.5
3. Solve the equation. Write a reason for each step.
3 = 5
4. Solve for x and write a reason for each statement.
Substitution
Subtraction
Subtraction
3
Class Notes 2.5
Reflexive Property of Equality
Real Numbers:
For any real number a, a = a.
Segment Length:
For any segment , AB = AB.
Angle measure:
∠
∠
For any angle A, m A = m A.
∠
Symmetric Property of Equality
Real Numbers:
For any real numbers a and b, if a = b, then
b = a.
Segment Length:
For any segments CD = AB.
Angle Measure:
, if AB = CD, then For any angles A and B, if m A = m B, ∠
∠
∠
∠
∠
then m B = m A.
∠
Transitive Property of Equality
Real Numbers:
For any real numbers a, b, anc c, if a = b and b = c, then a = c.
Segment Length:
For any segments , ,
if AB = CD and CD = EF, then AB = EF.
Angle Measure:
For any angles A, B, and C, if ∠
∠ ∠
m A = m B and m B = m C, then ∠
∠
∠
∠
∠
m A = m C.
∠
4
Class Notes 2.5
Use the property to complete the statement.
a. Multiplication Property of Equality: If RS = TU, then x(RS) = _______________.
b. Division Property of Equality:
∠
∠
If 3(m 1) = m 2, then m 1 = ____________.
∠
c. Transitive Property of Equality:
If 2 = bc and bc = de, then ____________.
d. Substitution Property of Equality: If x = 3c and r = 5x + 7, then ______________.
5. Using the diagram, show that CF = AD.
Equation
Reason
(From the diagram)
5
Class Notes 2.5
∠
∠
6. In the diagram, m ABD = m CBE.
∠
∠
Show that m 1 = m 3.
Equation
Reason
Substitution
Subtraction
7. Show that the area of square ABCD is equal to the area of square EFGH.
6
Class Notes 2.5
Homework
108-111 (4 - 34 even)
Supplemental Problems
7