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Questions and Answers
Match the following theorems with their descriptions:
Exterior Angle Theorem (EAT) = The interior angles of any triangle have a sum of 180 degrees Angle Sum of a Triangle Theorem = Any exterior angle of a triangle is equal to the sum of the opposite interior angles Opposite Angle Theorem (OAT) = When two straight lines cross, opposite angles are equal. Exponent Property = None of the above
Match the following algebraic processes with their descriptions:
Cross Multiplication = Simplification of polynomial-based fractions by dividing like factors Partial Fraction Decomposition = Evaluating an algebraic expression by breaking down a rational expression into simpler forms Inverse of Multiplication = Dividing from top and bottom of a fraction Combining Like Fractions = Simplifying fractions by adding or subtracting like terms
Match the following logarithmic properties with their descriptions:
Log Property of Multiplication = log a(xy) = log a(x) + log a(y) Log Property of Division = log a(x/y) = log a(x) - log a(y) Change of Base Formula = log a(b) = log c(b) / log c(a) Log Property of Exponentiation = log a(x^n) = n log a(x)
Match the following pre-algebra properties with their descriptions:
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Match the following trigonometric identities with their descriptions:
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Match the following algebraic techniques with their descriptions:
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Match the following geometric shapes with their respective formulas for the area:
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Match the following algebraic identities with their respective expansions:
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Match the following trigonometric functions with their respective definitions:
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Match the following calculus concepts with their respective formulas:
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Match the following algebraic expressions with their respective factorizations:
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Match the following geometric formulas with their respective shapes:
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Match the algebraic identity with its expansion:
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Match the type of fraction with its description:
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Match the theorem with its statement:
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Match the factorization with its expression:
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Match the polynomial characteristic with its description:
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Match the procedure with its application:
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Study Notes
Geometry Theorems and Properties
- Exterior Angle Theorem: Any exterior angle of a triangle equals the sum of the opposite interior angles.
- Angle Sum of a Triangle: The sum of the interior angles of any triangle is always 180 degrees.
- Opposite Angle Theorem: When two straight lines intersect, opposite angles are equal.
Algebraic Operations
- Order of Operations: Remember using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
- Simplification: Utilize techniques like combining like terms, cross-multiplying, and factoring.
- Complete The Square: A method for factoring quadratic expressions, also helpful for solving equations.
Properties of Logarithms
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Logarithmic Identities:
- logₐ(xy) = logₐ(x) + logₐ(y)
- logₐ(x/y) = logₐ(x) - logₐ(y)
- logₐ(x^n) = n * logₐ(x)
Basic Algebra Properties
- Commutative Property: Order of addition or multiplication does not affect the result (a + b = b + a, ab = ba).
- Associative Property: Grouping of numbers does not affect the sum or product (a + (b + c) = (a + b) + c).
- Distributive Property: a(b + c) = ab + ac, relates multiplication to addition.
Polynomial Operations
- Rational Zero Theorem: Rational zeros of a polynomial take the form r/s, where r is a factor of the constant term and s is a factor of the leading coefficient.
- Factoring by Grouping: Group terms to factor polynomials and simplify expressions.
Geometry Formulas
- Right Triangle: Apply the Pythagorean theorem (c² = a² + b²) for calculating side lengths.
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Area of Shapes:
- Parallelogram: A = base * height
- Trapezoid: A = (base1 + base2) * height / 2
- Triangle: A = (1/2) * base * height
- Volume Formulas: For cones and cylinders, use V = (1/3)πr²h for cones and V = πr²h for cylinders.
Trigonometry
- Definition of Trigonometric Functions: Ensure knowledge of sine, cosine, and tangent based on right triangle definitions.
- Proper and Improper Fractions: Cross-multiply for improper fractions, perform polynomial long division if necessary.
Fundamental Theorem of Algebra
- An nth-degree polynomial has exactly n roots (not necessarily distinct); an odd-degree real polynomial must have at least one real root.
Special Factoring Techniques
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Special Factors:
- Difference of squares: x² - a² = (x - a)(x + a)
- Sum or difference of cubes: x³ ± a³ = (x ± a)(x² ∓ ax + a²)
Binomial Theorem
- Expands powers of binomials:
- (x + y)² = x² + 2xy + y²
- (x - y)³ = x³ - 3x²y + 3xy² - y³
These notes summarize essential concepts, formulas, and theorems useful for studying mathematics across geometry, algebra, and trigonometry.
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Description
Test your knowledge of geometry formulas and theorems, including the Exterior Angle Theorem, Angle Sum of a Triangle Theorem, and Opposite Angle Theorem. Review and learn key concepts and formulas to improve your geometry skills.