Geometry Formulas and Theorems

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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:

Commutative Property of Addition = a + b = b + a Associative Property of Multiplication = a × (b × c) = (a × b) × c Distributive Property = a × (b + c) = a × b + a × c Identity Property of Addition = a + 0 = a Signup and view all the answers

Match the following trigonometric identities with their descriptions:

Pythagorean Identity = sin^2(x) + cos^2(x) = 1 Reciprocal Identity = sin(x) = 1 / csc(x) Quotient Identity = tan(x) = sin(x) / cos(x) Cofunction Identity = sin(x) = cos(π/2 - x) Signup and view all the answers

Match the following algebraic techniques with their descriptions:

Factoring = Expressing an expression as a product of simpler expressions Complete the Square = Solving quadratic equations by adjusting the form of the equation Inverse of Addition = Subtracting the same value from both sides of an equation Inverse of Multiplication = Dividing both sides of an equation by a value Signup and view all the answers

Match the following geometric shapes with their respective formulas for the area:

Circle = A = πr² Trapezoid = A = (a + b)h/2 Right Triangle = A = (b × h)/2 Parallelogram = A = bh Signup and view all the answers

Match the following algebraic identities with their respective expansions:

(x + y)² = x² + 2xy + y² (x - y)² = x² - 2xy + y² (x + y)³ = x³ + 3x²y + 3xy² + y³ (x - y)³ = x³ - 3x²y + 3xy² - y³ Signup and view all the answers

Match the following trigonometric functions with their respective definitions:

sin(θ) = opposite side / hypotenuse cos(θ) = adjacent side / hypotenuse tan(θ) = opposite side / adjacent side csc(θ) = hypotenuse / opposite side Signup and view all the answers

Match the following calculus concepts with their respective formulas:

Volume of a Right Circular Cone = V = (1/3)πr²h Surface Area of a Sphere = SA = 4πr² Volume of a Cylinder = V = πr²h Lateral Surface Area of a Cone = LSA = πr√(r² + h²) Signup and view all the answers

Match the following algebraic expressions with their respective factorizations:

acx³ + adx² + bcx + bd = (ax² + b)(cx + d) a(b + c) = ab + ac a² + 2ab + b² = (a + b)² a³ - b³ = (a - b)(a² + ab + b²) Signup and view all the answers

Match the following geometric formulas with their respective shapes:

c² = a² + b² - 2abcos(θ) = Triangle (Law of Cosines) h = arcsin( opposite side / hypotenuse) = Right Triangle A = (a + b)h/2 = Trapezoid V = (1/3)πr²h = Right Circular Cone Signup and view all the answers

Match the algebraic identity with its expansion:

(x + y)^2 = x^2 + 2xy + y^2 (x - y)^2 = x^2 - 2xy + y^2 (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 Signup and view all the answers

Match the type of fraction with its description:

Improper fraction = The degree of the numerator is higher than the degree of the denominator Proper fraction = The degree of the numerator is smaller than the degree of the denominator Mixed fraction = A combination of a whole number and a proper fraction Rational expression = A fraction where the numerator and denominator are polynomials Signup and view all the answers

Match the theorem with its statement:

Fundamental Theorem of Algebra = An nth degree polynomial has n (not necessarily distinct) zeros Quadratic Formula = If p(x) = ax^2 + bx + c, and 0 ≤ b^2 - 4ac, then the real zeros of p are x = (-b ± √(b^2 - 4ac))/2a Binomial Theorem = (x + y)^n = x^n + nx^(n-1)y + ... + y^n Remainder Theorem = If p(x) = a_nx^n + ... + a_1x + a_0, then p(a) = a_n(a)^n + ... + a_1a + a_0 Signup and view all the answers

Match the factorization with its expression:

x^2 - a^2 = (x - a)(x + a) x^3 + a^3 = (x + a)(x^2 - ax + a^2) x^4 - a^4 = (x^2 - a^2)(x^2 + a^2) x^3 - a^3 = (x - a)(x^2 + ax + a^2) Signup and view all the answers

Match the polynomial characteristic with its description:

Zero of a polynomial = A value of x that makes the polynomial equal to zero Factor of a polynomial = An expression that divides the polynomial evenly Root of a polynomial equation = A value of x that satisfies the equation p(x) = 0 Solution of a polynomial equation = A value of x that satisfies the equation p(x) = 0 Signup and view all the answers

Match the procedure with its application:

Partial fraction decomposition = Solving improper fractions that can't be separated into a proper fraction and an improper fraction Long division = Simplifying an improper fraction by dividing the numerator by the denominator Cross multiplication = Simplifying a fraction by canceling out common factors Synthetic division = Evaluating a polynomial at a specific value of x Signup and view all the answers

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

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.
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

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.