Points, Lines, Angles and Circles

• A point is a location in space, lacking size or dimension.
• A line is a straight, one-dimensional figure extending infinitely in both directions, defined by length but without width.
• A line segment is a finite portion of a line, bounded by two endpoints.
• A ray is a line that originates from a specific point and extends infinitely in a single direction.
• A circle is the set of all points equidistant from a central point. This distance is known as the radius.

Angles

• An angle is formed by two rays sharing a common endpoint, known as the vertex.
• An acute angle measures less than 90°.
• A right angle measures exactly 90°.
• An obtuse angle measures greater than 90° but less than 180°.
• A straight angle measures exactly 180°.

Supplementary and Complementary Angles

• Complementary angles are two angles that sum to 90°.
• Supplementary angles are two angles that sum to 180°.

Conversion Between Percentages and Decimals

• To convert a percentage to a decimal, divide it by 100, which effectively moves the decimal point two places to the left. For example, 75% becomes 0.75.
• To convert a decimal to a percentage, multiply it by 100, moving the decimal point two places to the right. For example, 0.4 becomes 40%.

Percentage, Base, and Rate

• The formula to find a percentage is: Part (Percentage) = Base × Rate
• Base refers to the whole amount.
• Rate is the percentage expressed in decimal form.
• Percentage is the portion of the base being considered.
• Example: 20% of 50 = 50 × 0.20 = 10

Properties of Real Numbers

• Commutative Property of Addition: a + b = b + a
• Commutative Property of Multiplication: ab = ba
• Associative Property of Addition: (a + b) + c = a + (b + c)
• Associative Property of Multiplication: (ab)c = a(bc)
• Distributive Property: a(b + c) = ab + ac
• Additive Identity Property: a + 0 = a
• Multiplicative Identity Property: a × 1 = a
• Additive Inverse Property: a + (-a) = 0
• Multiplicative Inverse Property: a × (1/a) = 1, where a ≠ 0

Operations on Integers

• For addition: When adding integers with the same sign, add their values and keep the sign. When adding integers with different signs, subtract their values and keep the sign of the larger number.
• For subtraction: Change the subtraction to addition and flip the sign of the second number. Example: 5 − (−3) = 5 + 3 = 8
• For multiplication and division: Multiplying or dividing integers with the same sign yields a positive result. Multiplying or dividing integers with different signs yields a negative result.

Algebraic Expressions

• An algebraic expression consists of variables, numbers, and operations. Example: 3x + 2
• Terms are the parts of an algebraic expression separated by + or − signs.
• A coefficient is a number multiplied by a variable such as 3 in 3x.
• A variable is a symbol, usually a letter, representing an unknown number.

Polynomials

• A polynomial is an expression containing multiple terms. Example: 2x^2 + 3x − 5
• The degree of a polynomial is the highest exponent of its variable.
• A monomial contains one term (e.g., 4x).
• A binomial contains two terms (e.g., x + 3).
• A trinomial contains three terms (e.g., x^2 + 2x + 1).
• Polynomials can be added or subtracted by combining like terms.
• Polynomials are multiplied using the distributive property, or the FOIL method for binomials.