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Algebra II. Lessons 1-5. LESSON 1. Polygons- Triangles Transversals Proportional Segments. Polygon. A polygon is a closed geometric figure located in one plane (flat surface) whose sides are: Straight line segments Intersect only at endpoints. Convex vs. concave polygons.

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algebra ii

Algebra II

Lessons 1-5

lesson 1

LESSON 1

Polygons-

Triangles

Transversals

Proportional Segments

polygon
Polygon
  • A polygon is a closed geometric figure located in one plane (flat surface) whose sides are:
    • Straight line segments
    • Intersect only at endpoints
slide4

Convex vs. concave polygons

Convex polygons can have all interior points connected without ever going outside the shape.

In concave polygons it is possible to have to go outside the shape to connect interior points.

regular polygons
REGULAR POLYGONS
  • A regular polygon is defined as a polygon which has all sides congruent and all angles congruent.
    • SQUARE
    • EQUILATERAL TRIANGLE
slide7

A DIAGONAL of any polygon is defined as a line through the interior of the polygon which connects two of the vertices.

How many other diagonals does this pentagon have?

triangles
TRIANGLES

By angle:

angle sum theory
Angle Sum Theory

For a convex polygon, the sum of the interior angles equals the following:

Where n-equals the number of sides of the polygon

Sum of interior angles in degrees = (180) X (n-2)

equilateral triangle
EQUILATERAL TRIANGLE

All the sides are congruent

Therefore

All the angles are congruent (60 degrees each)

isosceles triangle
ISOSCELES TRIANGLE

In a triangle – if two sides are congruent then the angles opposite those side must also be congruent.

20

20

50

50

transversals and parallel lines
TRANSVERSALS AND PARALLEL LINES

If these lines are parallel they will never intersect.

This is a transversal through the parallel lines.

Let’s see what happens when there are parallel lines cut by a transversal.

proportional segments
PROPORTIONAL SEGMENTS

3

2

9/2

3

When 3 or more parallel lines are cut by two transversals, the lengths of corresponding segments of the transversals are proportional.

algebra ii1

ALGEBRA II

Lesson 2

Negative Exponents

Product and Power Theorems for Exponents

Circle Relationships

zero exponent theorem
ZERO EXPONENT THEOREM
  • Any non-zero number (or variable) taken to the zero power is equal to - - -

1

negative exponents
NEGATIVE EXPONENTS
  • Definition of
    • If n is any real number and x is any real number that is NOT zero:
negative exponents cont
Negative Exponents (cont)
  • When we write an exponential expression in reciprocalform, the sign of the exponent must be changed.
    • If exponent is negative, it is changed to positive in the reciprocal form
    • If exponent if positive, it is changed to negative in the reciprocal form.
product theorem for exponents
Product theorem for Exponents
  • If m and n and x are real numbers and
slide25

Why????

Wow, that was easy!

power theorem for exponents
Power Theorem for Exponents
  • If m and n and x are real numbers.

EXTENSION OF POWER THEOREM:

algebra ii2

ALGEBRA II

Lesson 3

Evaluation of Expressions

Adding Like Terms

expressions
EXPRESSIONS
  • Numerical expressions: an arrangement of numerals and symbols
  • Algebraic expressions: can contain variables that represent unknown numbers.
evaluating expressions
Evaluating Expressions

EVALUATE:

Let x = -2 and y = -4

adding like terms
Adding Like Terms

LIKE terms: terms whose literal components represent the same number regardless of the numbers used to replace the variables.

What????

Simply put: terms whose variables are alike.

slide35

3xyz

-2zyx

LIKE TERMS

SIMPLIFY by adding LIKE terms:

3xy – 2x + 4 – 6yx + 3x

algebra ii3

ALGEBRA II

Lesson 4

Distributive Property

Solutions of Equations

Change Sides – Change Signs

distributive property
DISTRIBUTIVE PROPERTY
  • Definition:
    • a(b+c) = ab + ac

Demonstrate.

equations
Equations
  • Demonstration.
solutions of equations
Solutions of Equations
  • From Algebra I the five steps to solving an equation are
  • Eliminate the parenthesis.
  • Add like terms on both sides.
  • Eliminate the variable on one side or the other. (I don’t like to deal with negatives)
  • Eliminate the constant term on the side with the variable.
  • Eliminate the coefficient of the variable.
change sides change signs
Change Sides – Change Signs
  • Transposition: Change sides – change signs.

Change sides – change signs

algebra ii4

ALGEBRA II

Lesson 5

Word Problems

Fractional Parts of a number

slide47

Example 5.1: Twice a number is decreased by 7, and this quantity is multiplied by 3. The result is 9 less than 10 times the number. What is the number?

Highlight the IMPORTANT stuff.

Twice decreased by quantity multiplied

result less than 10 times

Define the variable (unknown). N= number

Write in algebraic terms

2n -7 3( )

= -9 10( )

Write the equation and solve:

3(2n-7) = 10n-9

slide48

Example 5.2: The number of ducks on the pond was doubled when the new flock landed. Then, 7 more ducks came. The resulting number was 13 less than 3 times the original number. How many ducks were there to begin with?

Highlight:

Define:

Algebraic terms:

Equation & Solve:

fractional parts of a number
Fractional Parts of a Number

Finding a part (fraction) of a number means to multiply the number times the fraction. (A lot like percent).

slide50

Example 5.4: One fifth of the clowns had red noses. If 30 clowns had red noses, how many clowns were there in all?

Highlight:

Define:

Algebraic terms:

Equation & Solve:

slide51

Seven eighths of the Tartar horde rode horses. If 140,000 were in the horde, how many did not ride horses?

Highlight:

Define:

Algebraic terms:

Equation & Solve: