Geometry

Angle Bisector Theorem: Statement, Proof, Examples

The angle bisector theorem states that the bisector of an angle in a triangle divides the opposite side into two segments whose ratio equals the ratio of the two adjacent sides: BD/DC = AB/AC. This article covers the statement, the internal and external versions, a clean similar-triangles proof, the converse, the length-of-bisector formula, six worked examples, the common mistakes, and where the theorem leads next.

Difference Between a Line and a Line Segment

The difference between a line and a line segment comes down to ends: a line has no endpoints and runs infinitely in both directions, while a line segment has two endpoints and a definite, measurable length. This article covers both definitions, a side-by-side comparison table, the notation, real-world examples, and the common mistakes students make telling them apart.

Cross Product of Two Vectors: Formula, Rule, Examples

The cross product of two vectors $\mathbf{a}$ and $\mathbf{b}$ produces a third vector perpendicular to both, written $\mathbf{a} \times \mathbf{b}$, with magnitude $|\mathbf{a}||\mathbf{b}|\sin\theta$ and direction set by the right-hand rule. This article covers the formula, the determinant method of computing it, the right-hand rule, the area-of-a-parallelogram link, and how it differs from the dot product.

Eccentricity of an Ellipse: Formula, Meaning, Examples

The eccentricity of an ellipse is a single number, $e = \dfrac{c}{a}$, that measures how stretched the ellipse is, from a perfect circle to a long, thin oval. This article covers what eccentricity measures, the two equivalent formulas $e = \dfrac{c}{a}$ and $e = \sqrt{1 - \dfrac{b^2}{a^2}}$, why it always lies between 0 and 1, the derivation, and worked examples.

Center of a Circle: Definition, Formula, How to Find

The center of a circle is the single fixed point that sits the same distance, the radius, from every point on the circle. This article covers the definition, the equation form $(x - h)^2 + (y - k)^2 = r^2$, and three ways to find the center: from the equation, from the endpoints of a diameter, and from three points on the circle.

Parallel Lines Cut by a Transversal: Angles, Properties, Examples

When two parallel lines are cut by a transversal, the crossing makes eight angles that fall into four named pairs — corresponding, alternate interior, alternate exterior, and co-interior angles — and the parallel condition forces each pair to be either equal or supplementary. This article maps the full configuration, defines all eight angles and every pair, lays out the properties, works six examples solving for x, and flags the common mistakes.

Obtuse Scalene Triangle: Properties & Examples

An obtuse scalene triangle is a triangle with one obtuse angle and all three sides of different lengths — combining the "obtuse" classification by angle with the "scalene" classification by side. This article covers the definition, the properties, how to find its area by base-and-height and by Heron's formula, real-world examples, six worked problems, and the common mistakes.

Angle Side Angle (ASA): Rule, Proof, Examples

The angle side angle (ASA) rule states that two triangles are congruent if two angles and the side included between them in one triangle equal the corresponding two angles and included side of the other — and the word included is what separates ASA from AAS. This article covers the statement, a full proof, two-column proof use, the ASA-versus-AAS difference, six worked examples, and the common mistakes.

Segment Addition Postulate: Formula & Examples

The segment addition postulate states that if point B lies between points A and C on a line segment, then AB + BC = AC — the two shorter parts add to the whole. This article covers the definition, the formula, why "between" matters, how it finds a midpoint, how it powers two-column proofs, six worked examples, and the mistakes to avoid.

Transitive Property of Congruence: Examples

The transitive property of congruence states that if one figure is congruent to a second and the second is congruent to a third, then the first is congruent to the third — in symbols, if a ≅ b and b ≅ c, then a ≅ c. This article covers the statement for segments, angles, and triangles, where it differs from the substitution property, two-column proofs that use it, six worked examples, and the mistakes to watch for.

What Is a Polyhedron? Types & Euler's Formula

A polyhedron is a three-dimensional solid bounded entirely by flat polygon faces, joined along straight edges that meet at points called vertices. This article defines the polyhedron, names its parts, walks through the types, and shows how Euler's formula, $F + V - E = 2$, ties faces, vertices, and edges together.

Irregular Polygons: Definition, Types & Area

Irregular polygons are closed flat shapes whose sides are not all equal and whose angles are not all equal — the opposite of regular polygons. This article defines them, lists the common types, shows that the interior-angle sum is still $(n-2)\times 180°$, and walks through finding the area by decomposition — splitting the shape into triangles and rectangles.

Exterior Angles of a Polygon: Sum & Formula

The exterior angles of a polygon are the angles between each side and the extension of its neighbour, and they always sum to $360°$ — for any polygon, regular or irregular, no matter how many sides. In a regular polygon, each exterior angle is $\dfrac{360°}{n}$. This article defines the exterior angle, derives the $360°$ sum, gives the formula, and works through examples.

Coterminal Angles: Definition, Formula, and Examples

Coterminal angles are angles in standard position that share the same terminal side — they land in the same place after differing by a whole number of full turns. You find them by adding or subtracting $360°$ (or $2\pi$ radians). This article covers the definition, the formula in degrees and radians, positive and negative coterminal angles, the unit-circle picture, and worked examples.

Obtuse Angle: Definition, Degrees & Examples

An obtuse angle is any angle that measures more than $90°$ and less than $180°$ — wider than a right angle but not yet a straight line. This article defines the obtuse angle, lists its properties, shows where it appears in real life and in triangles, and clears up the mistakes that trip students up.

X Intercept: Definition, Formula & Examples

Ordered Pair: Definition, (x, y) Notation & Examples

An ordered pair is two numbers written $(x, y)$ in a fixed order that together name a single point, where the order matters: $(2, 4)$ and $(4, 2)$ are different points. This article covers the notation, how to plot an ordered pair, the equality rule, where order changes the answer, and the mistakes students make most.

Coordinate Plane: Plot Points, Quadrants & Examples

A coordinate plane is the grid where every point is located by an ordered pair $(x, y)$ — the x-coordinate first, the y-coordinate second. This article shows how to plot a point, how to read a point's coordinates off a graph, the four quadrants and their sign rules, and the mistakes students make most.

Cartesian Plane: Definition, History & Quadrants

The Cartesian plane is the two-axis coordinate system invented by René Descartes that lets every point be named by an ordered pair $(x, y)$. This article covers who built it and why, how the plane is constructed from two perpendicular axes and an origin, the four quadrants, and worked examples of locating points.

Slope of Parallel Lines: Formula & Examples

The slope of parallel lines is the same for both lines: if two lines are parallel, then $m_1 = m_2$. This article explains why equal slopes force two lines to stay parallel, derives the rule, works through six examples, and clears up the mistakes that trip students up most.

Side Side Side (SSS): Congruence Proof & Examples

The side side side (SSS) rule states that if the three sides of one triangle equal the three sides of another, the triangles are congruent — with no angle information required at all. This article covers the statement, why three sides lock a triangle's shape, the proof, SSS similarity, six worked examples, the common mistakes, and where the rule leads next.

Points and Lines: Definition & Examples

Points and lines are the two most basic ideas in geometry: a point marks an exact position with no size, and a line is an endless, straight row of points. This article covers how a line is built from points, collinear versus non-collinear points, the incidence rules that connect them, and six worked examples.

Square: Properties, Area, Perimeter, Diagonal

A square is a quadrilateral with four equal sides and four right angles — the most regular four-sided shape there is. This article covers its properties and the three formulas, all derived from the shape itself: area $A = s^2$, perimeter $P = 4s$, and diagonal $d = s\sqrt{2}$, with six worked examples and the mistakes students make most.

Straight Line: Definition, Properties & Slope

A straight line is a one-dimensional figure that extends infinitely in both directions, has no curves, and keeps a constant slope throughout. This article covers the definition and properties, what slope means and how to find it, a brief tour of the equation forms, and six worked examples.

Scale in Maths: Scale Drawings and Map Scale

In maths, scale is the ratio between a length on a drawing or map and the matching length in real life, written like 1 cm : 5 km or 1 : 50,000. This article covers what scale means, how to read a map scale, how to convert between map distance and real distance in both directions, six worked examples, and the mistakes students make most.

Unit Circle: Definition, Coordinates, Chart, Examples

The unit circle is the circle of radius 1 centred at the origin, where every point on the rim has coordinates (cos θ, sin θ) for the angle θ measured from the positive x-axis. This article covers the definition, the equation x² + y² = 1, why the coordinates are cosine and sine, the special angles in both degrees and radians, six worked examples, and the common mistakes.

Rectangular Pyramid: Volume, Surface Area, Faces

A rectangular pyramid is a 3D solid with a rectangular base and four triangular faces that meet at a single apex — giving 5 faces, 8 edges, and 5 vertices. This article covers its properties, the volume formula $V = \tfrac{1}{3} \times l \times w \times h$, the surface area formulas, the net, six worked examples, and the mistakes students make most.

AAS Congruence Rule: Proofs, AAS vs ASA

The AAS congruence rule lets you prove two triangles congruent when two angles and a non-included side of one match the other — and the trick to using it is reading the figure to confirm the side sits outside the two angles. This article is a how-to-apply guide: when AAS is the right call, how to write the two-column proof, AAS versus ASA, six worked proof exercises, and the mistakes to avoid.

Hypotenuse Leg Theorem (HL): Proof & Examples

The hypotenuse leg theorem (HL) states that two right triangles are congruent if their hypotenuses are equal and one pair of legs is equal — just two matching sides, not three. This article covers the statement, why it works only for right triangles, the RHS link, the Pythagorean proof, six worked examples, the common mistakes, and where the rule leads next.

Reference Angle: Definition, Formulas, Examples

A reference angle is the positive acute angle between the terminal side of an angle and the x-axis, always between $0^\circ$ and $90^\circ$ ($0$ and $\tfrac{\pi}{2}$). This article covers the definition, the per-quadrant rules in both degrees and radians, how to handle negative and large angles, why reference angles let you evaluate trig functions anywhere, six worked examples, and the common mistakes.

Alternate Interior Angles Theorem: Proof

The alternate interior angles theorem states that when a transversal crosses two parallel lines, each pair of alternate interior angles is congruent (equal). This article gives the formal statement, a full two-step proof, the converse and its proof, the related co-interior angles theorem, and six worked examples. For the underlying definition of the angle pair, see Alternate Interior Angles.

Acute Scalene Triangle: Properties & Examples

An acute scalene triangle has all three angles less than 90° and all three sides of different lengths, so no two angles and no two sides ever match. This article covers the definition, how a triangle can be acute and scalene at once, the properties, the area and perimeter formulas with derivation, six worked examples, and the common mistakes.

Difference Between Rhombus and Rectangle

The difference between a rhombus and a rectangle is that a rhombus has four equal sides but slanted angles, while a rectangle has four right angles but only its opposite sides equal. This article covers both shapes' properties in full, what they share as parallelograms, the key differences in a comparison table, when each one matters, and six worked examples.

Radians to Degrees — Conversion Table and Formula

To convert radians to degrees, multiply by $180°/\pi$. The formula: $\text{degrees} = \text{radians} \times \dfrac{180°}{\pi}$. This article gives a complete conversion table for every common angle (multiples of $\pi/12$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$ and beyond), three worked examples, the reverse direction, and the most common mistakes.

Scale Factor: Definition, Formula & Examples

The scale factor is the number you multiply every length of a figure by to get the matching length of a similar figure, equal to new length ÷ original length. This article covers the formula, scaling up versus down, dilation on the coordinate plane, the area ($k^2$) and volume ($k^3$) rules, and six worked examples.

Dilation in Geometry: Definition, Scale Factor, Examples

A dilation in geometry is a transformation that resizes a figure about a fixed center of dilation by a scale factor, making it larger or smaller while keeping its shape exactly the same. This article covers the definition, the coordinate rule (x, y) → (kx, ky), how to find and use the scale factor, what happens with fractions and negative values, six worked examples, and the common mistakes.

Symmetry in Geometry - Types, Definition, Examples

Symmetry in geometry means a shape looks identical after being transformed — moved, rotated, or flipped. There are three core types: reflection symmetry (mirror image across a line), rotational symmetry (looks the same after rotation by a fixed angle), and point symmetry (every point has a matching point through a central point)

Geometric Transformations: Definition, Types and Examples

Foci of an Ellipse: Definition, Formula, Examples

The foci of an ellipse are two fixed points on its major axis such that, for any point on the ellipse, the sum of its distances to the two foci is constant (equal to 2a). This article covers the definition, where the foci sit, the formula c² = a² − b², six worked examples, and the common mistakes.

Ellipse - Equation, Formula, Properties, Graphing

An ellipse is the set of all points in a plane whose distances to two fixed points (called foci) sum to a constant. Its standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a$ is the semi-major axis and $b$ is the semi-minor axis.

Directrix of a Parabola: Definition, Equation, Examples

The directrix of a parabola is a fixed line, perpendicular to the axis, such that every point on the curve is exactly as far from the directrix as it is from the focus. This article covers the equidistance definition, the directrix equation for all four standard forms ($y^2 = 4ax$ gives directrix $x = -a$), how to find and derive it, six worked examples, and the common mistakes.

Focus of a Parabola: Definition, Formula, Examples

The focus of a parabola is the single fixed point on the axis such that every point on the curve is the same distance from the focus as it is from the directrix line. This article covers the definition, the focus formula for all four standard forms ($y^2 = 4ax$ gives focus $(a, 0)$), the focal distance, the latus rectum and focal chord, six worked examples, and the common mistakes.

Parabola - Definition, Formula, Graph, Examples

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). It's one of four classical conic sections — created by slicing a cone with a plane parallel to its slant side. The standard equation is $y^2 = 4ax$ (horizontal opening) or $(x - h)^2 = 4p(y - k)$ (vertex form).

Conic Sections: Types, Formulas & Equations

A conic section is the curve you get when a flat plane slices through a cone, and tilting the slice produces exactly four shapes: the circle, ellipse, parabola, and hyperbola. This article covers the definition, the four types, their eccentricity values, the focus-directrix idea, standard equations, six worked examples, and the mistakes students make most.

Parallel & Perpendicular Lines: Slope Rules

Parallel and perpendicular lines are two relationships you can read straight off slopes: parallel lines have equal slopes ($m_1 = m_2$), and perpendicular lines have slopes that multiply to −1 ($m_1 \cdot m_2 = -1$). This article covers both slope rules, why each works, how to tell lines apart from their equations, the special vertical-line case, and six worked examples.

Slope of Perpendicular Lines: The −1 Rule

The slope of perpendicular lines follows one rule: the product of the two slopes is −1, so $m_1 \cdot m_2 = -1$. This article covers the negative-reciprocal rule, why it holds, how to find a perpendicular slope from any given slope, the vertical-horizontal exception, and six worked examples.

Undefined Slope: Definition, Equation & Examples

An undefined slope is the slope of a vertical line, where every point shares the same x-coordinate, so the slope formula divides by zero. This article covers why a vertical line's slope is undefined, the equation x = a, the graph, how undefined slope differs from zero slope, and six worked examples.

Horizontal Line - Definition, Equation, and Slope

A horizontal line is a straight line that runs parallel to the x-axis. Its equation has the form $y = b$ (where $b$ is a constant), its slope is exactly $\mathbf{0}$, and it intersects the y-axis at the single point $(0, b)$.

Intercept Form of a Line: Formula & Examples

The intercept form of a line is $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the x-intercept (where the line crosses the x-axis) and $b$ is the y-intercept (where it crosses the y-axis). This article covers the formula, its derivation, how to read or graph a line straight from its intercepts, the triangle it cuts with the axes, and six worked examples.

Y Intercept: Definition, Formula & Examples

The y intercept is the point where a graph crosses the y-axis, found by setting $x = 0$ and solving for $y$. For a line written as $y = mx + b$, the y intercept is simply $b$. This article covers the definition, the method for every equation form, the parabola case, six worked examples, and the mistakes students make most.

y = mx + b: Read Slope & Y-Intercept, Plot a Line

In the equation $y = mx + b$, the number $m$ is the slope (how steeply the line rises or falls) and the number $b$ is the y-intercept (where the line crosses the y-axis). This article shows how to read $m$ and $b$ straight off the equation, plot the line, write the equation from points or a graph, and rearrange any linear equation into this form.

Slope Intercept Form: y = mx + b & Examples

Slope intercept form is the equation of a straight line written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This article covers how to read $m$ and $b$ straight off the equation, how to find them from points or a graph, how to convert other forms, and six worked examples.

Slope of a Line - Formula, Calculation, Examples

The slope of a line - sometimes called the gradient — measures the line's steepness as the ratio of vertical change to horizontal change between any two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$, or "rise over run."

Quadrants of Coordinate Plane - I, II, III, IV

The coordinate plane is divided by the x-axis and y-axis into four quadrants, numbered I, II, III, IV counterclockwise starting from the upper right. Each quadrant has a specific sign convention for $(x, y)$: Quadrant I: both positive; II: $x$ negative, $y$ positive; III: both negative; IV: $x$ positive, $y$ negative.

X and Y Axis in a Graph: Origin & Quadrants

In a graph, the x-axis is the horizontal number line and the y-axis is the vertical number line; they cross at the origin, the point (0, 0). This article explains the two axes, the four quadrants and their sign conventions, and how to plot any point from its ordered pair $(x, y)$, with six worked examples.

Triangular Pyramid: Volume, Surface Area & Faces

A triangular pyramid, also called a tetrahedron, is a 3D solid with 4 triangular faces, 6 edges, and 4 vertices — the simplest possible polyhedron. This article covers its properties, the volume formula $V = \tfrac{1}{3} \times \text{base area} \times \text{height}$, the surface area formulas, the net, six worked examples, and the mistakes students make most.

Triangular Prism - Volume, Surface Area, Formulas

A triangular prism is a 3D solid with 2 triangular bases and 3 rectangular lateral faces — total 5 faces, 9 edges, 6 vertices. The volume is $V = (\text{area of triangle}) \times L = \tfrac{1}{2}bh \times L$ where $b, h$ are the triangle's base and height, and $L$ is the prism's length. The surface area = sum of the two triangle areas + the three rectangle areas.

Rectangular Prism - Volume, Surface Area, Formulas

A rectangular prism (also called a cuboid) is a 3D solid with 6 rectangular faces, 12 edges, and 8 vertices. Its volume is $V = l \times w \times h$ (length × width × height), and its surface area is $S = 2(lw + lh + wh)$.

Cylinder — Shape, Formula, Examples

Central Angle in Geometry: Definition, Formula, Examples

A central angle in geometry is an angle whose vertex sits at the centre of a circle and whose two sides are radii, and its measure equals the measure of the arc it cuts off. This article covers the definition, the formula in both degrees and radians, the central angle theorem (the central angle is twice an inscribed angle on the same arc), six worked examples, and the common mistakes.

Tangent in Geometry — Definition, Formula, Examples

A tangent is a straight line that touches a curve at exactly one point and does not cross it there. This article covers the geometric tangent (focused on the circle), the formulas that describe a tangent line, the two foundational tangent theorems, three worked examples (Quick, Standard, Stretch), and the mistakes students make most often.

Equation of a Circle: Standard & General Form

The equation of a circle in standard form is (x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. This article covers the standard and general forms, the derivation straight from the Pythagorean theorem, how to read the centre and radius off either form, how to convert between them by completing the square, and six worked examples.

Arc Length: Formula, How to Find It, Examples

Arc length is the distance measured along the curved edge of a circle, a fraction of the full circumference set by the central angle: L = (θ/360°) × 2πr in degrees, or L = rθ in radians. This article covers the definition, both formulas, the derivation from the circumference, how to find arc length with and without the angle, six worked examples, and the common mistakes.

Sector of a Circle: Area, Arc Length, Perimeter

A sector of a circle is the pie-slice region enclosed by two radii and the arc between them, and its area is the fraction of the whole circle set by its central angle: Area = (θ/360°) × πr². This article covers the definition, minor and major sectors, the area, arc length, and perimeter formulas in both degrees and radians, six worked examples, and the common mistakes.

Area of a Circle: Formula, Derivation & Examples

The area of a circle is the flat space enclosed inside its boundary, given by the formula $A = \pi r^2$, where $r$ is the radius. This article defines the area, derives πr² by unrolling the circle into a triangle, covers area from the diameter and circumference, and works through six examples.

Circumference of a Circle - Formula, Examples

The circumference of a circle is the distance around it — its perimeter. Given the radius $r$, the formula is $C = 2\pi r$. Given the diameter $d = 2r$, equivalently $C = \pi d$. The constant $\pi \approx 3.14159$ is the ratio of any circle's circumference to its diameter — a universal property of all circles.

Chord of a Circle: Formula, Theorems, Examples

A chord of a circle is a straight line segment joining any two points on the circle's boundary, and the longest possible chord is the diameter. This article covers the definition, the two formulas for chord length (from the perpendicular distance and from the central angle), the main chord theorems with proof, six worked examples, and the common mistakes.

Diameter of a Circle — Formula, Worked Examples, and Properties

The diameter of a circle is the longest chord, passing through the centre, equal to twice the radius. Formula: $d = 2r$, or $d = C/\pi$ from circumference, or $d = 2\sqrt{A/\pi}$ from area. This article gives the three diameter formulas, three worked examples (Quick, Standard, Stretch), and the historical thread from Archimedes to modern usage.

Parts of a Circle: Names, Definitions & Diagram

The main parts of a circle are the centre, radius, diameter, circumference, chord, arc, sector, segment, tangent, and secant, every one of them defined by its relationship to the single fixed centre point. This article names and explains each part with a labelled diagram, the formulas tied to them, six worked examples, and the common mistakes students make.

Interior Angles: Sum Formula & Examples

Interior angles are the angles inside a polygon, one at each vertex. The sum of all of them is $(n-2) \times 180°$ for an $n$-sided polygon, and in a regular polygon each one equals that sum divided by $n$. This article covers the definition, the sum formula and where it comes from, regular versus irregular polygons, the interior angles between parallel lines, and six worked examples.

Hexagon Shape — Definition, Types, Properties, and Area Formula

Hexagon is a six-sided closed two-dimensional polygon with six vertices and six interior angles. In a regular hexagon, all six sides are equal, all six interior angles are $120°$, and the sum of interior angles is $720°$. The area of a regular hexagon with side $s$ is $\frac{3\sqrt{3}}{2}s^2$.

Pentagon Shape - Properties, Area, and Perimeter

A pentagon is a polygon with 5 sides and 5 interior angles summing to $540°$. A regular pentagon has all sides equal and all angles equal to $108°$ each. Its area formula is $A = \tfrac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \cdot s^2 \approx 1.72 s^2$, and its perimeter is $P = 5s$.

Shapes in Geometry — Complete 2D and 3D Taxonomy

Geometric shapes split into 2D (flat, with length and width) and 3D (solid, with length, width, and height). 2D shapes group into polygons (triangles, quadrilaterals, regular and irregular) and non-polygons (circle, ellipse). 3D shapes group into polyhedra (prisms, pyramids, Platonic solids) and curved solids (sphere, cylinder, cone).

Geometric Shapes: Types, Properties & Examples

Geometric shapes are closed figures built from points, lines, and curves, and they split into two families: flat 2D shapes and solid 3D shapes. This article covers the full list of types, the properties that separate one shape from the next, the area and volume formulas, six worked examples, and where shapes show up around you.

Difference Between Square and Rhombus Explained

The difference between a square and a rhombus is that a square has four right angles and equal-length diagonals, while a rhombus has four equal sides but its angles are not 90° and its diagonals are unequal. This article covers both shapes' properties in full, what they share, the key differences in a comparison table, when each one matters, and six worked examples.

Is a Square a Rectangle? Yes — Here's Why

Yes — every square is a rectangle, because a rectangle is defined as a quadrilateral with four right angles, and a square has those four right angles plus the extra condition that all its sides are equal. This article explains the definitions, the quadrilateral family tree, why a square is a special rectangle, why the reverse is not always true, six examples, and the common mistakes.

Properties of a Kite: Sides, Angles & Diagonals

A kite is a quadrilateral with two pairs of adjacent equal sides, diagonals that cross at right angles, and one pair of equal opposite angles. This article covers every property of a kite by sides, angles, diagonals, and symmetry, derives the area formula $\tfrac{1}{2} \times d_1 \times d_2$, and works through six examples.

Properties of a Rectangle: Sides, Angles & Diagonals

A rectangle is a quadrilateral with four right angles, opposite sides equal and parallel, and diagonals that are equal in length and bisect each other. This article covers every property of a rectangle by sides, angles, and diagonals, derives the area $l \times w$, perimeter $2(l+w)$, and diagonal $\sqrt{l^2 + w^2}$ formulas, and works through six examples.

Isosceles Trapezoid: Properties, Area & Examples

An isosceles trapezoid is a four-sided shape with one pair of parallel sides (the bases) and two non-parallel sides (the legs) of equal length, which gives it equal base angles, equal diagonals, and a line of symmetry. This article covers its definition, properties, the area formula $A = \tfrac{1}{2}(a+b)h$, perimeter, diagonals, six worked examples, and where students go wrong.

Trapezium - Definition, Properties, Area and Examples

A trapezium (US: trapezoid) is a quadrilateral with one pair of parallel sides. The parallel sides are called bases; the non-parallel sides are legs. The area formula is $A = \tfrac{1}{2}(a + b) \cdot h$ — the average of the parallel sides times the height.

Trapezoid: Properties, Area, and Formula Guide

Rhombus: Properties, Area, and Perimeter

Parallelogram - Properties, Area, and Formulas

Angle Angle Side (AAS) Congruence: Proof, Examples

Angle Angle Side (AAS) is a triangle congruence rule: if two angles and a non-included side of one triangle equal the corresponding two angles and side of another, the triangles are congruent. This article covers the statement, why it works, the proof from ASA, the difference between AAS and ASA, six worked examples, and the common mistakes.

CPCTC: Meaning, Proof & Examples

CPCTC stands for corresponding parts of congruent triangles are congruent: once two triangles are proven congruent, every matching pair of their sides and angles is automatically equal. This article covers what CPCTC means, why it follows from the definition of congruence, how it works as the final step of a two-column proof, six worked examples, the common mistakes, and where it leads next.

Congruent (Congruence) - Meaning, Definition, Examples

Congruent means identical in shape AND size. Two figures are congruent if one can be transformed into the other by rigid motions — translation, rotation, reflection — without stretching or shrinking. The symbol is $\cong$. For triangles, the five congruence theorems (SSS, SAS, ASA, AAS, RHS) let you prove congruence without measuring every side and angle.

Similar Triangles: Theorems & Properties

Similar triangles are triangles with the same shape but not necessarily the same size: their corresponding angles are equal and their corresponding sides are in the same ratio. This article covers the definition, the AA, SAS, and SSS similarity criteria, the properties, the area-ratio rule, the difference from congruent triangles, six worked examples, and the mistakes students make most.

Median of a Triangle: Properties & Formula

The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, so it always cuts that side into two equal halves. This article covers its definition, properties, the centroid and its 2:1 ratio, the length formula from Apollonius's theorem, six worked examples, and the mistakes students make most.

Altitude of a Triangle: Formulas & Properties

The altitude of a triangle is the perpendicular segment from a vertex straight down to the line containing the opposite side, and its length is the height used in the area formula. This article covers the definition, the formulas for scalene, isosceles, equilateral, and right triangles, the orthocentre, six worked examples, and the mistakes students make most.

Exterior Angle Theorem: Statement, Proof, Examples

The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two interior angles not next to it: exterior angle = sum of the two remote interior angles. This article covers the statement, a clean proof, the exterior angle inequality version, six worked examples, the common mistakes, and where the theorem leads.

Triangle Sum Theorem: Proof, Formula, Examples

The triangle sum theorem states that the three interior angles of any triangle always add up to 180° — written ∠A + ∠B + ∠C = 180°. This article covers the statement, the parallel-line proof, the exterior-angle link, why the rule holds only in flat (Euclidean) space, six worked examples, the common mistakes, and where the theorem leads next.

Pythagoras Theorem - Formula, Proof, Examples

Pythagoras theorem says a² + b² = c². Learn the formula, four proofs, common mistakes, the 4,000-year-old Babylonian tablet, and worked examples.

Right Scalene Triangle: Properties & Examples

A right scalene triangle has one right angle (90°) and three sides all of different lengths, which also forces its three angles to be different. This article covers the definition, how a triangle can be right and scalene at once, the properties, the area and perimeter formulas with derivation, six worked examples, and the common mistakes.

Right Angled Triangle: Properties & Formulas

A right angled triangle is a triangle with one angle of exactly 90°, and the side opposite that angle, the hypotenuse, is always the longest. This article covers its definition, properties, the Pythagorean theorem, area and perimeter formulas, the special 45-45-90 and 30-60-90 types, six worked examples, and the mistakes students make most.

Isosceles Obtuse Triangle: Properties & Examples

An isosceles obtuse triangle has one obtuse angle (between 90° and 180°) and two equal acute angles, with the two sides forming the obtuse angle equal in length. This article covers the definition, why such a triangle is possible, its properties, the area and perimeter formulas, six worked examples, and the common mistakes.

Isosceles Acute Triangle: Properties & Examples

An isosceles acute triangle has two equal sides, two equal angles, and all three angles less than 90°. This article covers the definition, why both labels can hold at once, its properties, the area and perimeter formulas, six worked examples, and the common mistakes students make.

Isosceles Right Triangle: Formulas & Examples

An isosceles right triangle is a right triangle whose two legs are equal, giving angles of 45°, 45°, and 90° and a fixed side ratio of 1 : 1 : √2. This article covers its definition, the 45-45-90 angle structure, area ($\tfrac{x^2}{2}$) and hypotenuse ($x\sqrt{2}$) formulas, six worked examples, and the mistakes students make most.

Isosceles Triangle - Definition, Types, Formulas

An isosceles triangle is a triangle with two sides of equal length — called the legs — and one side of different length called the base. The two angles opposite the equal sides (the base angles) are also equal.

Types of Triangles — Classification Matrix

Triangles are classified two ways — by **side lengths** (equilateral, isosceles, scalene) and by **angle measures** (acute, right, obtuse). Combining the two axes gives a $3 \times 3$ matrix with **seven** valid types and two impossible ones. This article gives the complete matrix, properties of each type, three worked examples, and the impossibilities that come from the triangle angle-sum theorem.

Consecutive Interior Angles: Theorem & Examples

Consecutive interior angles are the two non-adjacent interior angles that sit on the same side of a transversal, and the Consecutive Interior Angles Theorem says they are supplementary (sum to 180°) exactly when the two lines are parallel. This article covers the definition, the theorem, its converse for proving lines parallel, consecutive angles in a parallelogram, and six worked examples.

Same Side Interior Angles: Theorem & Examples

Same side interior angles are the two angles that lie between two lines and on the same side of the transversal, and when the two lines are parallel they are supplementary (their sum is 180°). This article covers the definition, the same side interior angles theorem and its converse, why they are supplementary, and six worked examples that solve for the supplementary angle.

Alternate Exterior Angles: Theorem & Examples

Alternate exterior angles are the pair of angles that lie outside two lines and on opposite sides of the transversal crossing them. When the two lines are parallel, each pair is equal. This article covers the definition, the theorem and its converse, how to spot the pairs, and six worked examples.

Alternate Interior Angles: Theorem & Examples

Alternate interior angles are the pair of angles that sit between two lines and on opposite sides of the line crossing them, called a transversal. When the two lines are parallel, each pair is equal. This article covers the definition, the theorem and its converse, how to spot the pairs, co-interior angles, and six worked examples.

Corresponding Angles — Postulate, Pair Table, Examples

Corresponding angles are pairs of angles that occupy the same relative position at each of the two intersections formed when a transversal crosses two lines. When the two lines are parallel, corresponding angles are equal.

Transversal — All 8 Angles and Pair Relationships

A transversal is a line that crosses two or more other lines at distinct points. When the transversal crosses two parallel lines, exactly $8$ angles form — grouped into four named pair-relationships (corresponding, alternate interior, alternate exterior, co-interior).

Angle Between Two Vectors: Formula & Examples

The angle between two vectors is found from their dot product: $\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}$, so $\theta = \cos^{-1}!\left(\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}\right)$. This article covers the formula, where it comes from, 2D and 3D worked examples, the cross-product alternative, six examples, and the common mistakes.

Angle Bisector: Properties & Construction

An angle bisector is a ray, line, or segment that divides an angle into two equal smaller angles. This article covers the definition, the key properties, the compass-and-straightedge construction, the angle bisector of a triangle and the incenter, and six worked examples.

Angle Addition Postulate: Formula & Examples

The angle addition postulate states that if a point $B$ lies in the interior of $\angle AOC$, then the two smaller angles add to the whole: $\angle AOB + \angle BOC = \angle AOC$. This article covers the definition, the formula, how to use it to solve for an unknown angle or for $x$, the link to straight and right angles, and six worked examples.

Congruent Angles: Definition, Theorems, Examples

Congruent angles are two or more angles that have the same measure, written $\angle A \cong \angle B$ using the congruence symbol ≅. This article covers the definition and symbol, the theorems that guarantee congruent angles, the compass-and-straightedge construction, and six worked examples.

Linear Pair of Angles: Definition & Axiom

A linear pair of angles is two adjacent angles whose non-common sides form a straight line, so they always add to 180°. This article covers the definition, the linear pair axiom and its converse, how a linear pair differs from supplementary and vertical angles, and six worked examples.

Supplementary vs Complementary Angles — A Side-by-Side Comparison

Supplementary angles add to $180°$; complementary angles add to $90°$. This article compares the two side by side, gives three worked examples (Quick, Standard, Stretch), explains the most common mistake, and offers the mnemonic that stops students mixing them up — Complementary forms a Corner, Supplementary forms a Straight line.

Supplementary Angles — Definition, Properties, Examples

Supplementary angles are any two angles whose measures sum to exactly 180°. The two angles can be next to each other (forming a straight line — a linear pair) or completely separate — what matters is the sum, not the position.

Complementary Angles — Definition, Properties, Examples

Complementary angles are any two angles whose measures sum to exactly 90°. The two angles can sit side by side (forming a right angle — a corner) or be drawn anywhere on the page — what matters is the sum. This article covers the definition, properties, the two types (adjacent vs non-adjacent), the right-triangle connection, three worked examples.

Vertical Angles — Definition, Theorem, Proof, and Examples

Vertical angles (also called vertically opposite angles) are the pair of non-adjacent angles formed when two straight lines cross at a single point. Sitting opposite each other across the intersection, they share only a vertex — never a side. The Vertical Angles Theorem states that vertical angles are always congruent (equal in measure), no matter how the two lines are oriented.

Adjacent Angles — Definition, Properties, and Examples

Adjacent angles are two angles that share a common vertex, share a common side (arm), and do not overlap. They sit next to each other — the word adjacent comes from Latin adjacens, meaning "lying near". Adjacent angles can be any size; they don't have to add to a specific number.

What is Adjacent? Meaning, Adjacent Angles, Solved Examples

In geometry, adjacent means "next to each other" — sharing a common side, edge, or vertex. Adjacent angles share a vertex and a side but don't overlap. Adjacent sides in a polygon share a common vertex. Adjacent in a triangle (with respect to an angle) is the side touching the angle that isn't the hypotenuse.

Vertex Angle: Definition, Formula, Examples

The vertex angle is the angle formed between the two equal sides of an isosceles triangle, sitting opposite the base, while the two angles at the ends of the base are the equal base angles. This article covers the definition, the apex-versus-base distinction, the formula $\text{vertex} = 180^\circ - 2(\text{base angle})$, the polygon vertex angle, six worked examples, and the common mistakes.

90 Degree Angle: Definition & Construction

A 90 degree angle is a right angle: the exact quarter turn formed when two lines meet perpendicularly, marked with a small square instead of an arc. This article covers the definition, how to construct one with a compass and verify it with the 3-4-5 rule, where right angles hold up buildings, and six worked examples.

45 Degree Angle: Definition & Construction

A 45 degree angle is an acute angle that measures exactly half of a right angle (90° ÷ 2 = 45°), and it is the angle each leg makes with the hypotenuse in an isosceles right triangle. This article covers the definition, how to construct one with a compass and by paper folding, its trigonometric values, where it shows up, and six worked examples.

Types of Angles — Acute, Right, Obtuse, Reflex

An acute angle measures between $0°$ and $90°$ — but it is only one of five named angle types. This article gives a complete reference table for all five (acute, right, obtuse, straight, reflex), with definitions, diagrams, real-world examples, and the common mistakes students make when sorting them.

Types of Angles: Acute, Right, Obtuse, Reflex

The six types of angles by measure are acute (under 90°), right (exactly 90°), obtuse (90° to 180°), straight (180°), reflex (180° to 360°), and full (360°). This article defines each type with a diagram, then covers the angle-pair relationships — complementary, supplementary, adjacent, and vertical — and the common mistakes.

Plane Definition in Math: Meaning & Examples

In math, a plane is a perfectly flat, two-dimensional surface that extends infinitely in every direction and has no thickness. This article covers the plane definition in math, its properties, how three non-collinear points fix exactly one plane, how planes are named, the difference between parallel and intersecting planes, and six worked examples.

Collinear Points: Definition, How to Prove & Examples

Collinear points are three or more points that all lie on the same straight line. This article covers the definition, collinear versus non-collinear points, the three methods to prove collinearity (slope equality, zero triangle area, and the distance test), how each one works, and six worked examples.

Coplanar: Definition, Points, Lines & Examples

Coplanar means lying on the same flat plane: a set of points is coplanar if a single plane can contain all of them, and lines are coplanar if one plane holds both. This article covers the definition, coplanar versus collinear, coplanar and non-coplanar points and lines, how to test for coplanarity, and six worked examples.

Skew Lines: Definition, Distance & Examples

Skew lines are two straight lines in three-dimensional space that never intersect and are never parallel, because they lie in different planes. This article covers the definition, why skew lines exist only in 3D, how to spot them in a cube, the conditions that classify two lines, the distance formula, and six worked examples.

Congruent Lines: Definition, Properties & Examples

Congruent lines are really line segments of equal length, written $\overline{AB} \cong \overline{CD}$ using the congruence symbol $\cong$. This article explains why true lines cannot be congruent, the three properties of congruent segments, how to check congruence with a ruler or the distance formula, and six worked examples.

Segment Bisector: Definition, Types & Examples

A segment bisector is any point, line, ray, or segment that passes through the midpoint of a line segment and splits it into two equal halves. This article covers the definition, the four kinds of bisector, how the perpendicular bisector is the special $90°$ case, the midpoint formula for finding one, and six worked examples.

Perpendicular Bisector: Definition & Construction

A perpendicular bisector of a line segment is a line that cuts the segment into two equal halves and meets it at a right angle ($90°$). Every point on it is equidistant from the two endpoints. This article covers the definition, the properties, the equidistance theorem, the compass-and-straightedge construction, the circumcentre, and six worked examples.

Perpendicular Lines - Definition, Slope, Examples

Line Segment: Definition, Properties & Examples

A line segment is a part of a straight line bounded by two distinct endpoints, so it has a fixed, measurable length — unlike a line or a ray, which run on forever. This article covers the definition and notation, how a segment differs from a line and a ray, its properties, the distance formula for length, and six worked examples.

Lines in Geometry: Types & Examples

Lines in geometry are straight, one-dimensional paths that have no thickness and run on forever in both directions. This article sorts out the point–line–ray–segment family first, then walks through every type a student meets: horizontal, vertical, parallel, perpendicular, intersecting, transversal, and skew lines, each with a labelled diagram and worked examples.

Geometry — Concepts, Formulas, Shapes & Topics Guide

Geometry is the branch of mathematics that studies points, lines, angles, shapes, and space — how figures are built, measured, and related. This hub maps every geometry topic into ten clusters: foundations (points, lines, planes), angles, parallel lines and transversals, triangles, quadrilaterals and polygons, circles, 3D solids, coordinate geometry, conic sections, and transformations — each linking to a full guide with formulas and worked examples.