How to Build a 75 Degree Angle Out of Two Others
Learn the four angle pairs a transversal makes, find unknown angles across parallel lines, construct 60, 90, 30, 45, 75 and 105 degrees with compasses, and draw bisectors and parallels.
Read articleStudy notes · Free to read
Free study notes for CBSE & ICSE, Classes 6–12.
Chapter-by-chapter notes written straight from the curriculum. Read free, then practise the same chapter with an AI-generated quiz.
Board
Class
Latest posts
Published articles
Filter by board, class, and subject to find notes for exactly what you're studying.
An Obtuse Angle Has No Complement at All
Learn to tell a line from a segment and a ray, classify every type of angle, find complements and supplements, and use linear pairs and vertically opposite angles to calculate unknowns.
Read articleA Set With Three Elements Hides Eight Sets Inside It
Learn to use the subset and proper-subset symbols, list every subset and verify the two-to-the-n rule, find a complement against a universal set, and compute unions, intersections and differences.
Read articleA Set of Tall Students Is Not a Set at All
Learn what makes a collection well-defined, write sets in roster and set-builder form, count elements to classify sets as finite, empty or singleton, and tell equal sets from equivalent ones.
Read articleName the Unknown and the Problem Solves Itself
Learn to turn a sentence into an equation, solve problems on numbers, consecutive numbers and ages, handle coins and perimeter questions, and check your answer against the original wording.
Read articleMove a Term Across the Equals Sign and Its Sign Flips
Learn to tell an equation from an expression, solve by balancing and by transposition, handle brackets and unknowns on both sides, and clear fractional or decimal coefficients.
Read articleEvery Term Must Meet Every Other Term
Learn to multiply monomials and polynomials without missing a product, divide to recover one side of a rectangle from its area, clear algebraic fractions with the LCM, and remove nested brackets in order.
Read articleWhy 5x and 5x Squared Can Never Be Added Together
Learn to spot constants, variables, terms and coefficients, classify expressions and find their degree, collect like terms by row and column methods, and evaluate an expression by substitution.
Read articleOne Formula, Four Different Questions
Learn to identify principal, rate, time and amount, use the simple interest formula, handle time given in months or days, and rearrange the formula to find any missing quantity.
Read articleA Shop Can Give a Discount and Still Make a Profit
Learn to include overheads in cost price, calculate profit or loss percent, find the selling price from the cost price and back again, and work out discount, marked price and the profit left after it.
Read articleA Price Raised 20 Percent Does Not Come Back Down by 20
Learn to convert between percentages, fractions and decimals, find a percentage of a quantity, work backwards to the whole from a part, and calculate percentage increase and decrease correctly.
Read articleMore Workers Means Fewer Days, Not More
Learn to spot direct from inverse variation, solve both by the unitary method, handle men-and-days and provisions problems, and work out how long two taps take together.
Read articleA Ratio Between Paise and Rupees Must Be Fixed First
Learn to simplify ratios across different units, share a quantity in a given ratio, combine two ratios into one, test four numbers for proportion, and find the fourth, third and mean proportionals.
Read articleAnything Raised to the Power Zero Equals One
Learn to write numbers in exponential form and as prime factors, find the sign of a power with a negative base, apply all six laws of exponents, and handle negative indices.
Read articleYou Can Tell Whether a Fraction Will Ever Stop Dividing
Learn which fractions give terminating decimals and which recur, write recurring decimals with bar notation, turn a recurring decimal back into a fraction, and round to decimal places or significant figures.
Read articleYou Cannot Add Kilograms to Grams Until One of Them Changes
Learn decimal place value and how to convert to and from vulgar fractions, compare decimals safely, add and subtract across units of money, length and mass, and multiply or divide decimals.
Read articleHow to Find the Whole When You Only Know a Part
Learn to add and subtract mixed numbers, multiply and divide with reciprocals, read the word of correctly, simplify complex fractions, and work backwards from a part to the whole quantity.
Read articleA Fraction Can Have Another Fraction Hiding Inside It
Learn to classify fractions as decimal or vulgar, proper or improper, simple or complex, convert between improper and mixed forms, find common denominators, and squeeze a fraction between two others.
Read articleOne Number Has No Reciprocal at All
Learn to add and subtract rational numbers with unlike denominators, multiply and divide using the reciprocal, combine all four operations in the right order, and solve word problems.
Read articleWhy a Denominator Is Never Allowed to Be Zero
Learn what makes a number rational, reduce any rational number to standard form, build equivalent forms with a required numerator or denominator, plot negatives on a number line and order them.
Read articleThe Bar Over a Sum Must Be Cleared Before Anything Else
Learn the sign rules for multiplying and dividing integers, verify the six properties with real numbers, simplify nested brackets and the vinculum in the right order, and solve integer word problems.
Read articleSneezing at Dust Is Your Defence System Overreacting
Learn what an allergy and an allergen are, how histamine produces the symptoms, which common allergens cause which conditions, how to tell an allergy from an infection, and how skin tests work.
Read articleYour Hand Pulls Back Before Your Brain Knows Why
Learn the functions of the cerebrum, cerebellum and medulla, how the spinal cord is protected, how to trace a reflex arc step by step, and how reflex actions differ from voluntary ones.
Read articleThe Longest Cell in Your Body Runs From Hip to Toe
Learn why the body needs coordination, label every part of a neuron and its job, tell sensory from motor and relay neurons, and map the central, peripheral and autonomic nervous systems.
Read article