Concept of Vertically Opposite Angles

A Fraction represents a part of a whole or a ratio between two integers written in the form $\frac{a}{b}$ (where $b \neq 0$). Fractions are categorized as:

• Proper Fraction: Numerator is strictly less than denominator (e.g., $\frac{3}{7}, \frac{5}{8}$). Value is always $< 1$.
• Improper Fraction: Numerator is greater than or equal to denominator (e.g., $\frac{9}{4}, \frac{11}{5}$). Value is $\ge 1$.
• Mixed Fraction: Consists of a whole integer and a proper fraction (e.g., $2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}$).
• Reciprocal (Multiplicative Inverse): For any non-zero fraction $\frac{a}{b}$, its reciprocal is $\frac{b}{a}$, such that $\frac{a}{b} \times \frac{b}{a} = 1$.

When evaluating multi-operator mathematical expressions, calculation order is strictly governed by the VBODMAS hierarchy:

Always simplify nested expressions from the innermost bracket outwards: Vinculum $\rightarrow$ Round Brackets $( )$ $\rightarrow$ Curly Brackets $\{ \}$ $\rightarrow$ Square Brackets $[ ]$.

Critical Sign Rule: When removing a bracket preceded by a negative sign ($-$), every addition and subtraction sign inside that bracket must be reversed:
$-(x + y) = -x - y \quad \text{and} \quad -(x - y) = -x + y$.

Modern spreadsheet programs (Microsoft Excel, Google Sheets), algebraic programming compilers, and financial banking transaction engines strictly follow this deterministic VBODMAS sequence to prevent ambiguity in multi-million dollar computations.

A Ratio is a comparison of two or more quantities of the exact same kind and unit by division. Written as $a : b$ (read as "$a$ is to $b$"), where $a$ is the Antecedent (পূর্বপদ) and $b$ is the Consequent (উত্তরপদ).

• Dimensionless: A ratio has no units.
• Simplest Form: $a : b$ is in simplest form when $\text{HCF}(a, b) = 1$.
• Inverse Ratio: The inverse ratio of $a : b$ is $b : a$.
• Compound (Mixed) Ratio: For ratios $a : b$ and $c : d$, the compound ratio is $(a \times c) : (b \times d)$.

When given two separate ratios sharing a common variable, such as $A : B = a : b$ and $B : C = c : d$, we determine the Continued Ratio $A : B : C$ by equalizing the middle term $B$ using the LCM of $b$ and $c$:

$$\text{If } A : B = 2 : 3 \quad \text{and} \quad B : C = 4 : 5$$

Multiply $A : B$ by 4: $8 : 12$. Multiply $B : C$ by 3: $12 : 15$.
$$\mathbf{A : B : C = 8 : 12 : 15}$$

Four quantities $a, b, c, d$ are in Proportion if $a : b = c : d$, written as $a : b :: c : d$, where $a \times d = b \times c$ (Product of Extremes = Product of Means).

• Direct Proportion (সরল সমানুপাত): When an increase in one quantity causes a proportional increase in the other (e.g., more goods purchased $\rightarrow$ higher total cost). Ratio $\frac{x}{y} = k$ (constant).
• Inverse Proportion (ব্যস্ত সমানুপাত): When an increase in one quantity causes a proportional decrease in the other (e.g., higher vehicle speed $\rightarrow$ less time taken to travel a fixed distance; more workers $\rightarrow$ fewer days required). Product $x \times y = k$ (constant).

Percentage means "parts per hundred" (symbol $\%$, representing a denominator of 100).
$$\text{Fraction to } \% = \frac{a}{b} \times 100\% \qquad \% \text{ to Fraction} = \frac{x}{100}$$

Percentage Change Formula:
$$\text{Percentage Increase / Decrease} = \frac{|\text{New Value} - \text{Original Value}|}{\text{Original Value}} \times 100\%$$

In mixture problems, dividing a total quantity of solution (e.g., 60 litres of milk and water in ratio $3 : 2$) requires computing unit share: Total parts $= 3 + 2 = 5$. Milk $= \frac{3}{5} \times 60 = 36$ litres; Water $= \frac{2}{5} \times 60 = 24$ litres.

The Unitary Method is a universal mathematical technique wherein we first calculate the value of a single unit (by division or multiplication), and then determine the value of the required number of units.

• In direct variation: 1 unit value $=$ Total value $\div$ Number of units.
• In inverse variation: 1 worker requires $=$ Total workers $\times$ Days.

The physical relationship connecting distance traveled ($D$), uniform speed ($S$), and elapsed time ($T$) is:
$$\mathbf{Distance = Speed \times Time} \quad \Longleftrightarrow \quad S = \frac{D}{T} \quad \Longleftrightarrow \quad T = \frac{D}{S}$$

Speed is typically measured in kilometres per hour (km/h) or metres per second (m/s). Converting between them relies on exact conversion factors:

$$1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} = \mathbf{\frac{5}{18} \text{ m/s}} \qquad 1 \text{ m/s} = \frac{3600 \text{ m}}{1000 \text{ km}} = \mathbf{\frac{18}{5} \text{ km/h} = 3.6 \text{ km/h}}$$

Rule of Thumb: To convert from $\text{km/h} \rightarrow \text{m/s}$, multiply by $\frac{5}{18}$. To convert from $\text{m/s} \rightarrow \text{km/h}$, multiply by $\frac{18}{5}$.

When a moving train crosses a stationary object with negligible length (such as a telegraph post, an electric pole, a signal tree, or a standing person):
$$\mathbf{\text{Total Distance Covered} = \text{Length of the Train} (L_{\text{train}})}$$

$$\text{Time to cross} = \frac{L_{\text{train}}}{\text{Speed of Train}}$$

When a moving train crosses a stationary structure possessing substantial length (such as a railway platform, bridge, or tunnel of length $L_{\text{plat}}$), the train must travel its own length plus the length of the platform before its last carriage fully clears the structure:
$$\mathbf{\text{Total Distance Covered} = L_{\text{train}} + L_{\text{plat}}}$$

$$\text{Time taken} = \frac{L_{\text{train}} + L_{\text{plat}}}{\text{Speed of Train}}$$

An Algebraic Expression is formed by combining mathematical variables (like $x, y, z$) and constants using arithmetic operations. An expression composed of one term is a Monomial (e.g., $7x^2$); two terms is a Binomial ($3x + 5$); three terms is a Trinomial ($x^2 + 2x + 1$); and in general, a Polynomial.

Like Terms (সদৃশ পদ): Terms having identical variable factors with the same exponents (e.g., $5x^2y$ and $-3x^2y$). Only like terms can be directly added or subtracted.

Multiplying polynomials relies on the algebraic Distributive Law:
$$(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$$

When multiplying powers with the same base, add their exponents: $x^m \times x^n = x^{m+n}$.

The three foundational algebraic identities established in Class 7 and essential for Class 8 mastery are:

1. Square of a Binomial Sum: $$(a + b)^2 = a^2 + 2ab + b^2$$
2. Square of a Binomial Difference: $$(a - b)^2 = a^2 - 2ab + b^2$$
3. Difference of Two Squares: $$a^2 - b^2 = (a + b)(a - b)$$

By adding and subtracting the expansions of $(a + b)^2$ and $(a - b)^2$, we derive two high-frequency exam formulas:

• Adding the two expansions: $$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = \mathbf{2(a^2 + b^2)}$$ $$\implies a^2 + b^2 = \frac{(a + b)^2 + (a - b)^2}{2}$$
• Subtracting the two expansions: $$(a + b)^2 - (a - b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = \mathbf{4ab}$$ $$\implies ab = \left(\frac{a + b}{2}\right)^2 - \left(\frac{a - b}{2}\right)^2$$

A standard WBBSE examination question format involves reciprocal variable sums:

$$\left(x + \frac{1}{x}\right)^2 = x^2 + 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = x^2 + \frac{1}{x^2} + 2$$

$$\mathbf{x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2} \qquad \mathbf{x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2}$$

Consider a rectangular field of length $L$ and breadth $B$. A path of uniform width $w$ runs along its boundary:

• Case 1: Path OUTSIDE the field:
  Outer Length $= L + 2w$, Outer Breadth $= B + 2w$.
  $$\mathbf{\text{Area of Path} = [(L + 2w)(B + 2w)] - (L \times B)}$$
• Case 2: Path INSIDE the field:
  Inner Length $= L - 2w$, Inner Breadth $= B - 2w$.
  $$\mathbf{\text{Area of Path} = (L \times B) - [(L - 2w)(B - 2w)]}$$

Review of essential angle relationships on intersecting lines and transversals:

• Complementary Angles (পূরক কোণ): Two angles whose sum is exactly $90^\circ$.
• Supplementary Angles (সম্পূরক কোণ): Two angles whose sum is exactly $180^\circ$.
• Adjacent Angles (সন্নিহিত কোণ): Two angles sharing a common vertex and a common arm, with non-common arms on opposite sides. When their non-common arms form a straight line, they form a Linear Pair (sum $= 180^\circ$).
• Vertically Opposite Angles (বিপ্রতীপ কোণ): Formed by two intersecting lines; vertically opposite pairs are always equal in measure ($ngle 1 = ngle 3, ngle 2 = ngle 4$).

For any triangle $\Delta ABC$:

1. The sum of all three interior angles is always $180^\circ$: $\angle A + \angle B + \angle C = 180^\circ$.
2. Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of its two interior opposite angles: $\angle ACD = \angle A + \angle B$.
3. The sum of the lengths of any two sides of a triangle must be strictly greater than the third side: $a + b > c$.

In school term tests and board evaluations: (1) Always draw a neat, clear pencil diagram for geometry and path problems; (2) State the formula before substituting numeric values; (3) Explicitly specify units at every step (${\text{cm}}^2, \text{m}^2, \text{km/h}, \text{sec}$); (4) Double-check sign distribution when removing brackets; and (5) Box or highlight final numerical answers.