Trigonometry - Quia



Geometry Lesson Notes 2.6 Date ________________

Objective: Write algebraic proofs. Use properties of equality to write geometric proofs.

Properties of Equality for Real Numbers

For all numbers a, b, and c,

Reflexive Property a = a

Symmetric Property If a = b then b = a

Transitive Property If a = b and b = c, then a = c

Addition and

Subtraction Properties If a = b, then a + c = b + c

Multiplication and If a = b, then a(c = b(c and a/c = b/c if c ( 0

Division Properties

Substitution Property If a = b, then a may be replaced by b in any equation

or expression

Distributive Property a(b + c) = ab + ac

NOTE: We will be assuming the Commutative and Associative Properties of addition and

multiplication. No need to state them in a proof.

You must be able to recognize and use these properties!

You can use these properties to justify every step as you solve an equation. The group of

algebraic steps used to solve problems is called a deductive argument.

Example 1 (p 94): Verify Algebraic Relationships

Solve ½ (x + 16) = 5x − 1 for x and give a reason for each step.

Statement Reason

½ (x + 16) = 5x − 1 Given

x + 16 = 2(5x − 1) Multiplication / Substitution properties

x + 16 = 10x − 2 Distributive / Substitution properties

16 = 9x − 2 Subtraction / Substitution properties

18 = 9x Addition / Substitution properties

2 = x Division / Substitution properties

x = 2 Symmetric property

This deductive argument is an example of an algebraic proof of a conditional statement.

The conditional statement would be: If ½ (x + 16) = 5x − 1, then x = 2.

The hypothesis is the starting point (the given) of the proof.

The conclusion is the end of the proof, what we need to prove.

Listing the reasons (properties) for each step makes this a proof.

Two-column, or formal, proof: contains statements (the steps) and reasons

(the properties that justify each step) organized in two columns.

Example 2 (p 94): Write a Two-Column (Algebraic) Proof

Write a two-column proof of the following conditional statement.

If [pic], then [pic].

Given: [pic]

Prove: [pic]

Statements: Reasons

1. [pic] 1. Given

2. [pic] 2. Multiplication Property

3. [pic] 3. Distributive Property

4. [pic] 4. Addition Property

5. [pic] 5. Substitution Property (Combining Like Terms)

6. [pic] 6. Subtraction Property

7. [pic] 7. Substitution Property (Combining Like Terms)

8. [pic] 8. Division Property

9. [pic] 9. Substitution Property

Proofs in geometry are presented in the same manner. Algebra properties as well as definitions, postulates, and other true statements can be used as reasons in a geometric proof.

Since geometry also uses variables, numbers, and operations, we are able to use many of the properties of equality to prove geometric properties.

Segment measures and angle measures are real numbers, so we can use the properties of equality to describe relationships between segments and between angles.

Examples:

Property Segments Angles

Reflexive AB = AB m(C = m(C

Symmetric If XY = YZ, then YZ = XY If m(1 = m(2, then m(2 = m(1

Transitive If MN = NO and NO = OP, If m(K = m(L and m(L = m(M,

then MN = OP then m(K = m(M

Practice: Name the property of equality that justifies each statement.

Statement Property

If 5 = x, then x = 5 _______________________________

If ½ x = 9, then x = 18 _______________________________

If AB = 2x and AB = CD, then CD = 2x _______________________________

If 2AB = 2CD, then AB = CD _______________________________

Example 3 (p 96): Justify Geometric Relationships

If GH + JK = ST and [pic], then which of the following conclusions is true?

I. GH + JK = RP

II. PR = TS

III. GH + JK = ST + RP

A. I only B. I and II C. I and III D. I, II, and III

Example 4 (p 96): Geometric Proof

A starfish has five arms. If the length of arm 1 is 22 cm, and arm 1 is congruent to arm

2, and arm 2 is congruent to arm 3, prove that arm 3 has length of 22 cm.

Given: _____________________________

_____________________________

Prove: _____________________________

Proof:

Statements Reasons

1. ________________________ 1. ___________________________________________

2. ________________________ 2. ___________________________________________

3. ________________________ 3. ___________________________________________

4. ________________________ 4. ___________________________________________

( HW: A7a pp 97-100 #14-25, 29-31, 37-38

A7b 2-6 Skills Practice / Practice

-----------------------

E

22 cm

SC

E

D

C

B

A

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