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
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- practice b 6 rigby high school
- 6 3 conditions for parallelograms gateway high school
- lmn or rst abc mr walker
- denton independent school district home
- practice a triangle congruence cpctc
- reteach 6 4 properties of special parallelograms
- answer key conejo valley unified school district
- copyright © by holt rinehart and winston
- lesson mr mccutchen geometry classes
- name
Related searches
- trigonometry calculator triangle
- trigonometry calculator for right triangle
- right triangle trigonometry calculator
- trigonometry formulas for triangles
- trigonometry calculator sin cos tan
- trigonometry calculator degrees
- free trigonometry problem solver
- trigonometry inverse calculator
- trigonometry calculator online
- quia synonyms and antonyms
- quia antonyms rags to riches
- quia synonyms rags to riches