Core SAT Curriculum & Blueprints
Synchronized, lesson-by-lesson knowledge base covering all essential SAT Math concepts, formulas, dimensional analysis rules, factoring identities, volume reference sheet tips, extraneous radical warnings, and complete unit circle trigonometry.
Complete Curriculum with 3-Tier Difficulty Scaling
Every official domain progresses through 3 difficulty tiers: Foundations (Units 2–5), repeated with increased complexity in Medium (Units 6–9), and synthesized into Advanced / Hard (Units 10–13). Every lesson includes formulas, solving methods, difficulty comparisons, and Desmos bypass shortcuts.
Linear equations, inequalities, word problems, rate interpretation, graphs of lines, systems of equations, and inequality regions.
Lesson 1 of 8: Solving Linear Equations and Linear Inequalities
Core FoundationVariable isolation mechanics, negative multiplier sign flips, and single-variable constraint analysis.
ax + b = c ──► ax = c - b ──► x = (c - b) / aSubtract constant b from both sides, then divide by variable coefficient a.
-ax ≥ b ──► x ≤ -b / a (FLIP the sign!)Multiplying or dividing both sides by a negative number ALWAYS reverses the inequality symbol.
Distribute parentheses, combine like terms, move all variable terms to one side and constants to the other, then divide by the coefficient.
Simple integers with direct 1-step or 2-step isolation.
Equations containing multiple fractions or requiring common denominators.
Equations with unknown constants a and b where you determine constraints for which no real value of x exists (or infinitely many values exist).
Type left side into row 1 (y = Left) and right side into row 2 (y = Right). Click the intersection point; the x-coordinate is your exact answer without manual algebra.
Lesson 2 of 8: Linear Equation Word Problems
Word ProblemsTranslating real-world scenarios into y = mx + b, fixed fees, and variable rate equations.
y = m · x + bm = Rate per unit (variable change), b = Flat fee / starting amount (constant baseline at x = 0).
Identify what is constant (starting amount b) and what changes repeatedly with each unit (m · x). Set up the linear equation y = mx + b and solve for the requested quantity.
Clear, direct rates with easily identifiable starting values.
Budgets with limits and inequality thresholds.
Multi-tiered rates with piecewise changes across intervals.
Lesson 3 of 8: Linear Relationship Word Problems
Context InterpretationInterpreting slope rates of change and y-intercept constants in scientific and economic contexts.
m = Δy / Δx (Rate of Change)Amount that y increases or decreases for every 1-unit increase in x.
b = Value of y when x = 0Baseline, initial value, or starting measurement before any change occurs.
The SAT often gives a formula like H = 1.5t + 20 and asks: "What does the number 1.5 mean in this context?" Match m directly to the unit of x (e.g., increase in height per year). Match b to the baseline value when t = 0.
Identify the starting value (b) or constant rate (m) directly from a straightforward formula.
Calculate m from a table of values and describe its real-world meaning with units.
Interpret formulas with shifted arguments or grouped variables where the rate only applies after an initial threshold.
Lesson 4 of 8: Graphs of Linear Equations and Functions
Graph MechanicsSlope calculations, slope-intercept form, standard form conversions, and parallel/perpendicular lines.
m = (y₂ - y₁) / (x₂ - x₁) = Δy / ΔxRise over run between any two points on the line.
Ax + By = C ──► Slope = -A/B, y-int = C/B, x-int = C/AInstantly determine slope and intercepts without converting to y = mx + b by hand.
m₁ = m₂Lines have identical slopes and different y-intercepts.
m₂ = -1 / m₁ (Negative Reciprocal)Slopes multiply to -1 (e.g., -3/4 becomes +4/3).
Convert any given line into y = mx + b to immediately read the slope and intercept. For standard form Ax + By = C, use slope = -A/B.
Find the slope given two simple coordinate points.
Write the equation of a line perpendicular to a given line passing through a specific point.
A line passes through unknown coordinates with parameter variables and is perpendicular to another line; solve for the parameter.
Lesson 5 of 8: Solving Systems of Linear Equations
High FrequencyThe 3 solution types, ratio condition criteria, elimination, and substitution.
a₁ / a₂ ≠ b₁ / b₂Lines have different slopes and intersect at exactly 1 coordinate point (x, y).
a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂Lines have identical slopes but different y-intercepts (parallel lines that never meet).
a₁ / a₂ = b₁ / b₂ = c₁ / c₂Lines have identical slopes and identical y-intercepts (the same exact line).
Use elimination by multiplying one row to match coefficients, or substitution if a variable is already isolated. For "no solution" or "infinite solutions" questions, set up coefficient ratios.
Add or subtract two equations directly to eliminate a variable.
Multiply both rows by different integers to eliminate a term with fractional solutions.
Find the unknown constant k that results in "no solution" or "infinitely many solutions".
Type both equations into Desmos. Click the gray intersection point to read (x, y) immediately. For unknown k, add a slider for k and adjust until lines become parallel.
Lesson 6 of 8: Systems of Linear Equations Word Problems
Word ProblemsQuantity vs. value equations, secondary question traps, and mixture problems.
x + y = Total ItemsRepresents total headcount, tickets, coins, or units.
(Price₁ · x) + (Price₂ · y) = Total CostWeights each variable by its monetary value, weight, or percentage.
Set up the quantity equation first, then the value equation. Solve for the requested variable using elimination or substitution.
Direct setup of standard tickets or coin counts.
Solve for variables, then answer a secondary question (e.g., "What is 3x - y?" or "How many more adults than children?").
Word problems involving percentage mixtures and chemical solutions.
Lesson 7 of 8: Linear Inequality Word Problems
Constraint ModelingTranslating English constraint keywords (at least, at most, exceeds) into mathematical inequalities.
"At least", "Minimum", "No less than" ──► ≥Must meet or exceed the threshold value.
"At most", "Maximum", "No more than" ──► ≤Must stay under or equal to the ceiling value.
"Greater than", "Exceeds" ──► > | "Fewer than", "Less than" ──► <Strict inequality where equality is not permitted.
Identify constraints in the word problem and match them to the correct inequality symbols. Pay close attention to boundary conditions ("inclusive" vs "strict").
Translate a single constraint directly into an algebraic inequality.
Two simultaneous constraints combining hours and earnings.
Identify which coordinate point (x, y) satisfies both constraints simultaneously under edge-case boundary conditions.
Lesson 8 of 8: Graphs of Linear Systems and Inequalities
Feasible RegionsDashed vs. solid boundary lines, half-plane shading, and polygonal feasible regions.
< or > ──► Dashed line | ≤ or ≥ ──► Solid lineDashed line means points on line are NOT solutions; solid line means points on line ARE solutions.
y ≥ mx + b ──► Shade ABOVE | y ≤ mx + b ──► Shade BELOWShade the half-plane that satisfies the inequality. Origin test (0, 0) verifies correct region.
Graph the boundary lines, determine shading for each inequality, and locate the overlapping intersection region (feasible set).
Identify which quadrant or region represents a single inequality.
Find the overlapping intersection region of two linear inequalities.
Given a system of 3 inequalities forming a shaded polygon, find coordinates of its vertices or test whether a specific point is inside the region.
Ratios, unit conversions, percentages, center and spread, scatterplots, linear vs. exponential, probability, data inferences, and statistical claims.
Lesson 1 of 10: Ratios, Rates, and Proportions
ProportionsCross-multiplication, unit rate conversions, part-to-part vs. part-to-whole ratios, and chained ratios.
a / b = c / d ──► a · d = b · cSet cross-products equal to solve for the missing variable.
Part / (Part₁ + Part₂)Divide individual part by the total sum of all parts.
Set up a clean proportion with matching units in the numerators and denominators, then cross-multiply to solve.
Simple cross-multiplication with whole numbers.
Unit rate conversions (e.g., cost per pound or miles per gallon) across mixed quantities.
Multi-variable chained ratios with shared intermediary variables.
Lesson 2 of 10: Unit Conversion & Dimensional Analysis
Dimensional FlowDiagonal unit cancellation, compound speed rates, and squared/cubed area and volume conversions.
(miles / hour) · (5280 ft / 1 mile) · (1 hour / 3600 sec)Place target units on the opposite side (top vs bottom) so unwanted units cancel out cleanly.
1 yd = 3 ft ──► 1 yd² = 9 ft² ──► 1 yd³ = 27 ft³When converting square units, square the conversion ratio; for cubic units, cube the ratio!
Chain conversion factors as multiplying fractions so every unwanted unit appears in both numerator and denominator, leaving only target units.
Single-step conversions like minutes to hours or inches to feet.
Compound rate conversions changing both numerator and denominator simultaneously.
Area and volume unit conversions requiring squaring or cubing the conversion ratio.
Lesson 3 of 10: Percentages, Percent Change & Multipliers
High FrequencyDirect percentage multipliers, percent change formulas, sequential shifts, and reverse solving.
Part = (Percent / 100) · WholeDirect multiplication of decimal percent by whole base.
% Change = ( |New - Old| / Old ) · 100%Always divide the difference by the ORIGINAL (Old) baseline value.
Increase by x% ──► × (1 + x/100) | Decrease by x% ──► × (1 - x/100)Increase by 15% ──► multiply by 1.15. Decrease by 20% ──► multiply by 0.80.
Use decimal multipliers directly rather than calculating the percent and adding/subtracting in two separate steps.
Calculate a direct single percentage of a number.
Consecutive sequential percent changes.
Reverse percentage solving where final discounted price is given and original price is requested.
Lesson 4 of 10: Center, Spread, and Shape of Distributions
StatisticsMean vs. median, outlier pull, range, and standard deviation spread comparisons.
x̄ = (Sum of all values) / (Total count n)Arithmetic average of all data points.
Middle number when ordered from least to greatestIf n is even, average the two middle numbers.
Mean is heavily pulled by outliers; Median is resistantAn extreme high value drags the mean up, while the median barely changes.
Measures spread/dispersion of data around the meanHigher dispersion/spread ──► Larger σ. Tightly clustered data ──► Smaller σ.
Order values from smallest to largest to find the median. For spread questions, inspect how far data values are clustered away from the center.
Calculate mean, median, and range of 5 given numbers.
Identify which of two visual bar graphs or dot plots has a larger standard deviation based on visual spread.
Explain how adding an extreme outlier changes the mean versus the median, or analyzing linear transformations on σ.
Lesson 5 of 10: Data Representations (Tables & Plots)
Data LiteracyReading and analyzing bar charts, histograms, frequency tables, and dot plots.
Mean = Σ(Value · Frequency) / Σ(Frequency)Multiply each value by its count, sum them up, and divide by total counts.
Check axis labels, bin widths on histograms, and scale increments before performing calculations.
Read values directly from bar chart or histogram axes.
Calculate the median or range from a dot plot or frequency table.
Combine two frequency tables or determine missing frequencies given a combined weighted average.
Lesson 6 of 10: Scatterplots, Trend Lines & Residuals
Regression AnalysisLines of best fit, positive/negative correlation, and calculating residuals.
y = m x + bm = predicted rate of change, b = predicted initial value at x = 0.
Residual = Actual y - Predicted yVertical distance from the data point to the trend line (positive if point is above line, negative if below).
Substitute the given x-value into the trend line equation to find the predicted y. Compare with the actual point coordinate.
Identify whether correlation is positive, negative, or zero.
Use the line of best fit equation to predict y for a given x-value.
Find the data point with the greatest absolute residual (furthest vertical distance from the trend line).
Lesson 7 of 10: Linear and Exponential Growth
Growth ModelsConstant addition vs. constant percentage multiplication, and hybrid rate scenarios.
y = mx + b (Adds/Subtracts Constant)Changes by a constant fixed amount per unit of time (e.g., adds 5 items each year).
y = a · (b)ᵗ (Multiplies by Constant Factor)Multiplies by a constant ratio or percentage per unit of time (e.g., grows by 5% each year).
Inspect whether differences (y₂ - y₁) are constant (Linear) or ratios (y₂ / y₁) are constant (Exponential).
Classify a verbal description as linear or exponential.
Write the functional model from a table of values.
Word problems that switch between linear and exponential rates over different time intervals.
Lesson 8 of 10: Probability and Relative Frequency
ProbabilityTwo-way contingency tables, conditional probability restrictions, and algebraic cell solving.
P = Favorable Outcomes / Total OutcomesRatio of target outcomes to total possible outcomes in sample space.
P(A given B) = Count of (A and B) / Total Count of BCrucial: Restrict the denominator ONLY to the group specified after "given that" or "of the".
Highlight the group specified in the prompt. That group count is your denominator. The overlap is your numerator.
Simple single-variable probability from a two-way table.
Conditional probability restricting the denominator to a specific row or column.
Fill in missing table cells algebraically using marginal totals before calculating the probability.
Lesson 9 of 10: Data Inferences & Margin of Error
InferencePopulation proportion estimation, sample size relationships, and margin of error confidence intervals.
Estimated Total = Sample Proportion · Total PopulationMultiply sample rate by the entire population size.
Confidence Interval = Estimate ± Margin of ErrorThe true population parameter is expected to fall between (Estimate - MOE) and (Estimate + MOE).
Larger Sample Size (n) ──► Smaller Margin of ErrorIncreasing sample size reduces uncertainty and narrows the margin of error.
Results from a sample can only be generalized to the population if the sample was chosen randomly. Check sample size effects on confidence precision.
Estimate a population total from a sample proportion.
Interpret margin of error intervals to determine plausible population values.
Determine whether increasing sample size reduces the margin of error or evaluating whether two confidence intervals overlap significantly.
Lesson 10 of 10: Evaluating Statistical Claims & Study Design
Study DesignRandom selection vs. random assignment, correlation vs. causation, and confounding bias.
Random Selection ──► Generalize to Entire PopulationAllows generalizing findings to the broader population from which subjects were sampled.
Random Assignment (Treatment vs Control) ──► Proves CausationProves a cause-and-effect relationship. Observational studies prove correlation only!
Look for two keywords: "Random Selection" (generalize to population) and "Random Assignment" (cause and effect). If no random assignment, eliminate choices claiming causation.
Identify obvious sample bias in survey setups.
Distinguish between correlation and causation in survey claims.
Identify confounding variables or undercoverage bias in experimental study setups.
Factoring, exponent rules, polynomial operations, rational expressions, nonlinear functions, quadratic formula, linear-quadratic systems, and graph multiplicity.
Lesson 1 of 13: Factoring Quadratic and Polynomial Expressions
Core FactoringDifference of squares, perfect square trinomials, grouping, and higher-degree factoring.
a² - b² = (a - b)(a + b)Two perfect squares subtracted factor into conjugate binomials.
a² ± 2ab + b² = (a ± b)²Square of first, twice the product of both, square of last.
ax² + bx + c ──► Split middle term into two factorsFind two numbers that multiply to a·c and add to b.
Look for a common factor (GCF) first. Then inspect for difference of squares or split the middle term for quadratics.
Simple factoring where leading coefficient a = 1.
Factoring quadratics where leading coefficient a > 1.
Higher-degree difference of squares and nested polynomial factors.
Lesson 2 of 13: Radicals and Rational Exponents
Exponent LawsThe 6 core laws of exponents, rational powers, and common base rewriting.
xᵃ · xᵇ = xᵃ⁺ᵇ | xᵃ / xᵇ = xᵃ⁻ᵇAdd exponents when multiplying; subtract when dividing.
(xᵃ)ᵇ = xᵃᵇMultiply exponents when raising a power to another power.
x⁻ᵃ = 1 / xᵃInvert the base into the denominator.
xᵃᐟᵇ = ᵇ√(xᵃ) = (ᵇ√x)ᵃNumerator = power, Denominator = root index.
x⁰ = 1 (for x ≠ 0)Any non-zero base to the 0 power equals 1.
Convert all radicals to fractional exponents to combine using exponent laws. Rewrite different bases into common prime bases (e.g. 4 and 8 into base 2).
Rewrite rational exponents as radicals or vice versa.
Simplify expressions combining negative and fractional powers.
Solve exponent equations with different bases by converting to a common prime base.
Lesson 3 of 13: Operations with Polynomials
Algebraic OperationsPolynomial addition, subtraction, FOIL expansion, and coefficient matching.
axⁿ + bxⁿ = (a + b)xⁿOnly terms with the exact same variable and exponent can be combined.
(a + b)(c + d) = ac + ad + bc + bdDistribute every term in the first polynomial across every term in the second.
Distribute negative signs across parentheses carefully. For coefficient matching, equate corresponding degree terms on both sides.
Add or subtract simple polynomials.
Multiply a binomial by a trinomial.
Match polynomial coefficients on both sides of an identity equation to solve for constants.
Lesson 4 of 13: Operations with Rational Expressions
Rational FunctionsFactoring rational fractions, polynomial long division, and the Remainder Theorem.
P(x) / Q(x) ──► Factor both, cancel common factorsOnly common FACTORS can be cancelled, never individual terms!
P(x) / D(x) = Q(x) + R(x) / D(x)Quotient plus Remainder over Divisor.
Remainder of P(x) ÷ (x - a) equals P(a)Plug in x = a directly to find the remainder without performing long division.
Factor numerator and denominator completely before simplifying. For improper rational fractions, use polynomial long division or synthetic division.
Simplify a basic rational fraction by factoring.
Add or subtract rational expressions with different denominators.
Use polynomial long division to rewrite rational expressions into quotient and remainder form, or apply Remainder Theorem.
Lesson 5 of 13: Nonlinear Functions & Composition
Function MechanicsFunction notation, composite functions (f ∘ g)(x), and solving for unknown inputs.
f(a) ──► Replace every x with aSubstitute the input value into every occurrence of the variable.
(f ∘ g)(x) = f(g(x))Evaluate inside function g(x) first, then plug that output into outside function f.
Work from the inside out for composite functions: calculate g(x) first, then apply f to that result.
Evaluate f(a) for a given numerical input.
Evaluate composite functions f(g(x)) using given formulas, tables, or graphs.
Solve for an unknown input x such that a composite function equals a target value.
Lesson 6 of 13: Isolating Quantities (Literal Equations)
Formula RearrangementRearranging multi-variable formulas, clearing denominators, and factoring out target variables.
Isolate target variable by reversing operations in PEMDAS orderClear denominators first, group all terms containing the target variable on one side, factor it out, and divide.
Multiply through by common denominators. Collect all terms containing the target variable on one side, factor out the variable, then divide by the remaining expression.
Rearrange a simple formula for one variable.
Solve for a variable inside a radical or squared term.
Solve for target variable when it appears in both numerator and denominator.
Lesson 7 of 13: Solving Quadratic Equations & The Discriminant
High FrequencyThe quadratic formula, square root method, and discriminant (b² - 4ac) root analysis.
x = [ -b ± √(b² - 4ac) ] / (2a)Solves any quadratic equation in standard form ax² + bx + c = 0.
D = b² - 4acD > 0 ──► 2 Distinct Real Solutions | D = 0 ──► 1 Real Solution | D < 0 ──► 0 Real Solutions (2 Complex)
Set the equation equal to 0. Identify a, b, and c. Compute D = b² - 4ac to check the nature of roots before applying the quadratic formula.
Solve by taking the square root directly.
Solve equations requiring the quadratic formula with radical solutions.
Find the unknown constant k such that the quadratic has exactly one real solution.
Lesson 8 of 13: Linear and Quadratic Systems
System SynthesisSubstitution between lines and parabolas, intersection points, and tangency constraints.
Set mx + b = ax² + bx + c ──► ax² + (b - m)x + (c - b) = 0Substitute line equation for y in the parabola equation to create a single quadratic.
Line tangent to parabola ──► Discriminant D = 0A tangent line touches the parabola at exactly ONE point.
Substitute the linear expression for y into the quadratic equation. Set equal to 0 and solve for x using factoring or quadratic formula.
Find intersection points graphically by inspecting a diagram.
Solve algebraically using substitution to find two intersection coordinates.
Find constant c such that a line is tangent to a parabola.
Lesson 9 of 13: Radical, Rational, and Absolute Value Equations
Trap WarningExtraneous roots, two-case absolute values, and zero-denominator domain restrictions.
|ax + b| = c ──► ax + b = c OR ax + b = -c (for c ≥ 0)Must split into two separate linear equations.
Squaring both sides can introduce false solutions: Check in original equation!When solving √(f(x)) = g(x), any solution making g(x) negative is extraneous.
Isolate the radical or absolute value first. Square both sides (for radicals) or branch into two cases (for absolute values). ALWAYS plug answers back into the original equation.
Solve simple absolute value equations directly.
Solve radical equations and eliminate the extraneous root.
Rational equations where an apparent algebraic solution creates a zero denominator.
Lesson 10 of 13: Quadratic and Exponential Word Problems
Word ProblemsProjectile trajectories, maximum vertex heights, and exponential time unit conversions.
h(t) = -16t² + v₀t + h₀h₀ = initial launch height, v₀ = initial vertical velocity. Time to max height = -v₀ / (2 · (-16)).
P(t) = P₀ · (b)ᵗP₀ = initial value at t = 0, b = growth factor (1 + r) or decay factor (1 - r).
For quadratic maximums, calculate the vertex x-coordinate x = -b/(2a). For exponentials, rewrite the exponent to match the desired time units.
Identify the initial launch height or starting population from the formula.
Find maximum height by calculating the vertex coordinates.
Convert annual growth models to monthly or daily models in the exponent.
Lesson 11 of 13: Quadratic Graphs & The Three Forms
Parabola AnatomyStandard, vertex, and factored forms, and which form reveals features as explicit constants.
y = ax² + bx + c ──► y-intercept is (0, c)Displays the y-intercept as an explicit constant.
y = a(x - h)² + k ──► Vertex is (h, k)Displays the maximum or minimum value (k) as an explicit constant.
y = a(x - r₁)(x - r₂) ──► x-intercepts are (r₁, 0) and (r₂, 0)Displays the roots / zeros as explicit constants.
Match the question request to the form: "reveals minimum/maximum" ──► Vertex form. "reveals x-intercepts" ──► Factored form. "reveals y-intercept" ──► Standard form.
Read the vertex directly from vertex form.
Match parabola graphs to their factored or vertex equations.
Questions asking which equivalent algebraic form reveals the vertex or intercepts as explicit constants without calculating.
Lesson 12 of 13: Exponential Graphs & Asymptotes
Asymptote AnalysisHorizontal asymptotes, y-intercepts, and horizontal/vertical curve transformations.
y = a · bˣ + ca = vertical stretch, b = growth/decay base, c = vertical shift.
y = cThe line the curve approaches as x approaches negative infinity (growth) or positive infinity (decay).
(0, a + c)Plug in x = 0: y = a · b⁰ + c = a(1) + c = a + c.
Look at the horizontal line the curve levels off toward; that horizontal line is y = c. Look at the y-intercept at x = 0.
Identify whether an exponential graph represents growth (b > 1) or decay (0 < b < 1).
Identify the horizontal asymptote from the equation.
Determine graph transformations under both horizontal and vertical shifts.
Lesson 13 of 13: Polynomial and Other Nonlinear Graphs
Multiplicity & BehaviorRoot multiplicity (cross vs. tangent turn), end behavior, and degree-4 polynomials.
(x - r)¹ ──► Graph CROSSES the x-axisChanges sign across the x-axis.
(x - r)² ──► Graph TOUCHES and TURNS AROUND (Tangent)Does not change sign; vertex is tangent to the x-axis.
Occur where denominator equals 0Values of x for which the rational function is undefined.
Identify where the graph crosses the x-axis (odd power root) versus where it touches and turns around (even power root). Write the factored equation and solve for stretch factor a using a given point.
Count real roots (x-intercepts) on a polynomial graph.
Match a cubic polynomial equation to its graph.
Determine the complete equation of a degree-4 polynomial from intercept tangencies and a given coordinate point.
3D volume formulas, dimensional scaling, similar triangles, right triangle trigonometry, circle proportions, unit circle coordinates, and circle equations.
Lesson 1 of 6: Area, Volume & Dimensional Scaling
3D Solid MechanicsCylinders, cones, spheres, pyramids, and the cubic dimensional scaling law.
V = π r² hCircular base area (πr²) multiplied by height h.
V = (1/3) π r² hExactly one-third the volume of a cylinder with identical dimensions.
V = (4/3) π r³Four-thirds times pi times radius cubed.
V = (1/3) B hB represents the area of the base polygon.
Lengths scale by k ──► Area scales by k² ──► Volume scales by k³When every linear dimension is multiplied by k, surface area scales by k² and volume scales by k³.
Identify the solid type. Substitute known dimensions into the reference formula. For scaling problems, apply k³ for volume and k² for surface area.
Direct volume calculation given radius and height.
Solve backwards for height or diameter given the volume.
Percent-based non-uniform dimensional changes across radius and height.
Lesson 2 of 6: Congruence, Similarity, and Angle Relationships
Similarity RatiosTriangle sum, transversal theorems, and similar triangle side-length proportions.
∠A + ∠B + ∠C = 180°The interior angles of any triangle always sum to 180°.
AB / DE = BC / EF = AC / DFCorresponding angles are EQUAL; corresponding side lengths are strictly PROPORTIONAL.
Alternate interior angles are equal; linear pairs sum to 180°Parallel lines cut by a transversal create pairs of equal alternate angles.
Match corresponding vertices between similar triangles. Set up side length ratios to solve for the missing segment.
Find supplementary linear pairs or simple interior triangle angles.
Solve for missing sides in nested similar triangles formed by parallel lines.
Rotated overlapping similar triangles with algebraic side lengths requiring quadratic solutions.
Lesson 3 of 6: Right Triangle Trigonometry & Special Triangles
Guaranteed 750+ TopicSOH-CAH-TOA, 4 core triples, 30°-60°-90° & 45°-45°-90° ratios, and the Complementary Angle identity.
sin θ = O / H | cos θ = A / H | tan θ = O / AO = Opposite, A = Adjacent, H = Hypotenuse.
3-4-5 | 5-12-13 | 8-15-17 | 7-24-25Multiples also form valid triples: 6-8-10, 9-12-15, 10-24-26, 30-40-50.
30°-60°-90°: x : x√3 : 2x | 45°-45°-90°: x : x : x√2Short leg = x (opposite 30°), long leg = x√3 (opposite 60°), hypotenuse = 2x.
sin(A) = cos(B) ──► A + B = 90° (or π/2 rad)The sine of an acute angle equals the cosine of its complementary angle.
For any right triangle, label Opposite, Adjacent, and Hypotenuse relative to angle θ. For complementary problems, set arguments to sum to 90° immediately.
Identify sin θ, cos θ, or tan θ directly from a labeled right triangle diagram.
Use special 30-60-90 or 45-45-90 ratios to find a missing altitude or hypotenuse.
Solve algebraic complementary angle equations with multi-step expressions.
Lesson 4 of 6: Circle Theorems (Arcs, Sectors, Angles)
Master ProportionThe 4-way circle proportion, radian arc/sector formulas, tangent perpendicularity, and semicircle theorems.
θ° / 360° = s / (2πr) = A / (πr²)Central angle over 360° equals arc length s over circumference (2πr) equals sector area A over total area (πr²).
Arc Length: s = r · θ | Sector Area: A = (1/2) · r² · θApplies exclusively when central angle θ is in radians.
Tangent meets radius at exactly 90°Forms a right angle with the radius at the point of contact.
Inscribed Angle = (1/2) · (Central Angle) | Angle facing diameter = 90°Any angle inscribed inside a semicircle is always a 90° right angle.
Set any two fractions in the 4-way proportion equal to each other to solve for the missing variable. Look for right triangles formed by tangent lines and radii.
Find circumference or full circle area given the radius.
Calculate sector area or arc length from a central angle in degrees.
Right triangles formed by tangent lines to circle radii, solved via the Pythagorean theorem.
Lesson 5 of 6: Unit Circle Trigonometry & Radians
Coordinate TrigRadian conversions, special angle benchmark values, and unit circle coordinates (cos θ, sin θ).
Radians = Degrees × (π / 180°) | Degrees = Radians × (180° / π)Multiply by π/180 to get radians; multiply by 180/π to get degrees.
x = cos θ | y = sin θ | tan θ = sin θ / cos θOn the unit circle, the x-coordinate is cos θ and the y-coordinate is sin θ.
30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2πKey angles to memorize for instant recall.
Determine the reference angle in Quadrant I, evaluate the trig value, then apply the quadrant sign (+ or -) based on coordinates.
Convert degrees to radians directly.
Evaluate trig values for special benchmark angles in Quadrant I.
Evaluate angles outside Quadrant I or negative rotations using reference angles and quadrant signs.
Lesson 6 of 6: Circle Equations in the xy-Plane
Step-by-Step BlueprintStandard form circle equations, center sign flip, radius determination, and completing the square.
(x - h)² + (y - k)² = r²Center is (h, k) [signs flip!], Radius = √r².
Add (b / 2)² to BOTH sides for both x and yTake half the middle coefficient and square it to create perfect square trinomials.
Group x-terms and y-terms. Move the constant to the right. Add (A/2)² and (B/2)² to both sides. Factor into (x - h)² + (y - k)² = r².
State center and radius directly from standard form.
Write the equation given center (h, k) and a point on the circumference.
Complete the square on expanded equations with x and y linear terms.
Type the expanded circle equation directly into Desmos. Click the leftmost edge and rightmost edge; the midpoint between them is the center (h, k), and half the distance between them is the radius r.