Search & glossary

Every rule and concept in Geometry, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (172)

A central angle equals its arc: A central angle has its vertex at the center of the circle.; Circles › Central Angles and Arc Measure
Add a line: When a figure gives you nothing to compute with, add an auxiliary line: a radius to a tangent point, an altitude, a diagonal.; Core Ideas in Plain Terms › Problem-Solving Playbook
Adjacent angles: Two angles are adjacent when they share a common vertex and a common side, but do not overlap (no interior points in common).; Angles › Adjacent Angles & the Angle Addition Postulate
An outside angle equals two inside angles: If you extend one side of a triangle, you create an exterior angle.; Triangles & Congruence › The Exterior Angle Theorem
Angle Addition Postulate: If point $B$ is in the interior of $\angle AOC$ , then the two small angles add up to the big angle: \[ m∠ AOB + m∠ BOC = m∠ AOC .; Angles › Adjacent Angles & the Angle Addition Postulate
Angle chasing: work one step at a time: Angle chasing means finding an unknown angle by filling in other angles one at a time, using the rules you already know, until the one you want is forced.; Angles › Going Deeper: Advanced Angle Ideas
Angle facts in triangles: Course Review › Triangles & Congruence
Angle pairs: Course Review › Angles & Angle Relationships
Angle sums in any polygon: For a polygon with $n$ sides: \[ sum of interior angles = (n-2)· 180^°, sum of exterior angles = 360^° .; Course Review › Quadrilaterals & Polygons
Angles formed by a transversal: When a transversal cuts two parallel lines, these pairs are formed:; Course Review › Parallel & Perpendicular Lines
Angles in a shape add to a fixed total: The Triangle Angle-Sum Theorem says the three angles of any triangle add to $180^\circ$ ; a quadrilateral adds to $360^\circ$ — both come from the polygon angle-sum formula $(n-2) \cdot 180^\circ$ .; Core Ideas in Plain Terms › Angles and Lines
Arc length: part of the way around: An arc is a fraction of the whole circle.; Circles › Circumference and Arc Length
Arcs: pieces of the circle itself: An arc is a curved piece of the circle between two points.; Circles › The Parts of a Circle
Area from coordinates: the shoelace formula: When you know the corner coordinates of a polygon, you can find its area directly — no base or height needed.; Quadrilaterals & Polygons › Going Deeper: Advanced Polygon Ideas
Area of 2D figures: Course Review › Perimeter, Area & Volume
Area of a circle: The area inside a circle is \[ A = π r^2 .; Circles › Area of a Circle and Area of a Sector
Area of a sector: A sector is a “pizza slice” bounded by two radii and an arc.; Circles › Area of a Circle and Area of a Sector
Area of a segment: A segment is the region between a chord and its arc — what is left of a sector after you remove the triangle formed by the two radii.; Circles › Going Deeper: Advanced Circle Ideas
Area Scales by $k^2$ , Volume Scales by $k^3$: When two figures are similar with scale factor $k$ (every length of the large figure is $k$ times the matching length of the small one), the effect compounds with each dimension:; Similarity › Going Deeper: Advanced Similarity
Base Angles Theorem: In an isosceles triangle, the two sides that are equal are the legs, and the third side is the base.; Triangles & Congruence › Isosceles and Equilateral Triangles
Beyond right triangles: Law of Sines & Law of Cosines: SOH-CAH-TOA only works in right triangles.; Right Triangles & Trig › Going Deeper: Advanced Right-Triangle & Trig Ideas
Bisectors and the angle between them: Angle bisectors set up equal pieces you can chain together.; Angles › Going Deeper: Advanced Angle Ideas
Break It Apart, Then Add or Subtract: A composite figure is made of familiar shapes joined together.; Area & Volume › Composite (Compound) Figures
Can these lengths make a triangle?: Three lengths form a triangle only if the two shorter ones can “reach across.” The rule:; Triangles & Congruence › The Triangle Inequality
Center–radius form: A circle with center $(h,k)$ and radius $r$ has the equation \[ (x-h)^2 + (y-k)^2 = r^2 .; Circles › The Equation of a Circle
Central, inscribed, and other angles: Course Review › Circles
Circumference, area, and arcs: With radius $r$ and diameter $d = 2r$ : \[ C = 2π r = π d, A = π r^2 .; Course Review › Circles
Circumference: the distance around: The circumference $C$ is the perimeter of a circle: \[ C = 2π r = π d .; Circles › Circumference and Arc Length
Clock hands as angles: A clock face is a circle of $360^\circ$ split into $12$ hours, so each hour mark is $360 / 12 = 30^\circ$ apart.; Angles › Going Deeper: Advanced Angle Ideas
Combined Solids — Deriving SA and Volume: A combined (composite) solid is glued from familiar solids: a cylinder capped by a hemisphere, an ice-cream cone (cone $+$ hemisphere), a house (prism $+$ pyramid), and so on.; Area & Volume › Going Deeper: Advanced Area & Volume
Composing Two Reflections: Reflecting twice in a row is always a single rigid motion.; Transformations › Going Deeper: Advanced Transformations
Congruence and Midpoint: Two segments are congruent ( $\overline{AB} \cong \overline{CD}$ ) when they have equal length ( $AB = CD$ ).; Points, Lines & Planes › Congruent Segments & Midpoints
Congruence proofs with an auxiliary line: Sometimes a proof seems stuck because there are not yet two triangles to compare.; Triangles & Congruence › Going Deeper: Advanced Triangle Ideas
Converse of the Pythagorean Theorem: Take the three sides of a triangle and let $c$ be the longest.; Right Triangles & Trig › The Converse: Right, Acute, or Obtuse?
Counting diagonals: From a single vertex you can draw $\mathbf{n-3}$ diagonals (you cannot connect a vertex to itself or to its two neighbors).; Quadrilaterals & Polygons › Polygon Vocabulary
Counting Lines, Planes, and Regions: When points and lines are in “general position” (no needless overlaps), the counts follow neat patterns.; Points, Lines & Planes › Going Deeper: Advanced Ideas
Cross-Sections, Nets, and the Scaling Laws: Cross-section: the flat shape you see when you slice straight through a solid.; Area & Volume › Going Deeper: Advanced Area & Volume
Cyclic quadrilaterals: opposite angles are supplementary: A cyclic quadrilateral is one whose four vertices all lie on a single circle.; Quadrilaterals & Polygons › Going Deeper: Advanced Polygon Ideas
Cyclic quadrilaterals: opposite angles are supplementary: A cyclic quadrilateral has all four vertices on a circle.; Circles › Going Deeper: Advanced Circle Ideas
Distance and midpoint are Pythagoras in disguise: The distance between two points is $\sqrt{(\Delta x)^2 + (\Delta y)^2}$ — the hypotenuse of the right triangle their coordinates make.; Core Ideas in Plain Terms › Going Deeper: Circles and Coordinates
Distance and Midpoint in Three Dimensions: A point in space needs three coordinates, $A(x_1,y_1,z_1)$ .; Points, Lines & Planes › Going Deeper: Advanced Ideas
Distance between two parallel lines: Parallel lines never meet, so they stay a fixed distance apart — but that distance is measured perpendicular to the lines, not vertically.; Parallel & Perpendicular Lines › Going Deeper: Advanced Ideas
Equation of a circle: A circle with center $(h,k)$ and radius $r$ : \[ (x-h)^2 + (y-k)^2 = r^2 .; Course Review › Circles
Extra powers: Each special parallelogram keeps all parallelogram properties and adds more:; Quadrilaterals & Polygons › Special Parallelograms: Rectangle, Rhombus, Square
Finding $n$ from an angle condition: Angle formulas run both directions: given a single interior or exterior angle of a regular polygon, you can solve for the number of sides $n$ .; Quadrilaterals & Polygons › Going Deeper: Advanced Polygon Ideas
Finding the center and radius by completing the square: An equation like $x^2 + y^2 - 6x + 4y - 12 = 0$ is a circle in disguise.; Circles › Going Deeper: Advanced Circle Ideas
Five kinds of angles: Angles › Classifying Angles
Five ways to prove congruence: You do not need to check all six parts (3 sides + 3 angles).; Triangles & Congruence › Triangle Congruence Shortcuts
Formula Reference — Flat Shapes: Let $b$ be a base and $h$ the matching perpendicular height.; Area & Volume › Perimeter & Area of Polygons
Formula Reference — Regular Polygons and Circles: Area & Volume › Regular Polygons & Circles
Formula Reference — Surface Area: Surface area (SA) is the total area of all the outside faces — the amount of wrapping paper needed.; Area & Volume › Surface Area of Solids
Formula Reference — Volume: Volume is how much space fills the inside — always in cubic units.; Area & Volume › Volume of Solids
Formulas on the Coordinate Plane: For points $A(x_1,y_1)$ and $B(x_2,y_2)$ : \[ Midpoint M = ( $\frac{x_1+x_2}{2}$ , $\frac{y_1+y_2}{2}$ ) \] \[ Distance AB = $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ \] The midpoint averages the coordinates.; Points, Lines & Planes › Midpoint Formula & Distance Formula
From Coordinates: the Shoelace Formula & Pick's Theorem: When a polygon is given by the coordinates of its corners, two tools find its area without any height at all.; Area & Volume › Going Deeper: Advanced Area & Volume
Glide Reflections: A glide reflection is a reflection over a line followed by a translation parallel to that line (the order does not matter).; Transformations › Going Deeper: Advanced Transformations
Heron's Formula — Area from the Three Sides: Sometimes you know a triangle's three sides but not its height.; Area & Volume › Going Deeper: Advanced Area & Volume
How Scaling Changes Size: If two similar figures have scale factor $k$ (each length of the big figure is $k$ times the matching length of the small one), then:; Similarity › Scale Factor: Perimeter and Area
Indirect Measurement with Two Unknowns: Sometimes a single similar-triangle setup hides two unknowns — for example an unknown height and an unknown distance.; Similarity › Going Deeper: Advanced Similarity
Inscribed angles are half the central angle: An angle drawn from the circle's edge sees an arc at half the angle drawn from the center.; Core Ideas in Plain Terms › Going Deeper: Circles and Coordinates
Isosceles triangle & the triangle inequality: Course Review › Triangles & Congruence
Justifying always / sometimes / never: The family tree turns every classification claim into a logic question.; Quadrilaterals & Polygons › Going Deeper: Advanced Polygon Ideas
Linear pairs: When two angles are adjacent and their outer sides form a straight line, they make a linear pair.; Angles › Linear Pairs & Vertical Angles
Looking up and looking down: An angle of elevation is measured up from the horizontal to an object above you.; Right Triangles & Trig › Angles of Elevation & Depression
Measuring the Unreachable: You can find a height you cannot reach by using similar triangles.; Similarity › Indirect Measurement (Shadows & Mirrors)
Name the four pairs: Corresponding angles sit in the same corner at each crossing (one up, one down).; Parallel & Perpendicular Lines › The Eight Angles and Their Pairs
Naming the pieces: A circle is the set of all points that are the same distance from one point, the center.; Circles › The Parts of a Circle
Nested and Overlapping Similar Triangles: Similar triangles often share a vertex or side, so they overlap on the page.; Similarity › Going Deeper: Advanced Similarity
Parallelogram properties: In every parallelogram: opposite sides are parallel and congruent; opposite angles are congruent; consecutive angles are supplementary; and the diagonals bisect each other.; Course Review › Quadrilaterals & Polygons
Perpendicular means right angles: When two lines are perpendicular ( $\ell \perp m$ ), they cross to make four $90^\circ$ angles.; Parallel & Perpendicular Lines › Perpendicular Lines and Right Angles
Power of a Point: From a point, the products of the two chord/secant/tangent pieces are equal:; Course Review › Circles
Preview: the identity $\sin^2\theta + \cos^2\theta = 1$: In a right triangle with hypotenuse $c$ , opposite side $a$ , and adjacent side $b$ , we have $\sin\theta = a/c$ and $\cos\theta = b/c$ .; Right Triangles & Trig › Going Deeper: Advanced Right-Triangle & Trig Ideas
Proportions, midsegment, and scaling: Course Review › Similarity
Proving the Inscribed Angle Theorem: Why is an inscribed angle exactly half its arc?; Circles › Going Deeper: Advanced Circle Ideas
Proving triangles congruent: Two triangles are congruent if any one of these matches:; Course Review › Triangles & Congruence
Pythagorean Theorem and its converse: In a right triangle with legs $a,b$ and hypotenuse $c$ (the side opposite the right angle): \[ a^2 + b^2 = c^2 .; Course Review › Right Triangles & Trigonometry
Reading a proof: Do not read a proof the way you read a rule.; Transformations › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Points, Lines & Planes › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Angles › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Parallel & Perpendicular Lines › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Triangles & Congruence › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Similarity › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Right Triangles & Trig › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Quadrilaterals & Polygons › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Circles › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Area & Volume › Why It Works
Reflection for Shortest Paths: To find the shortest path from point $A$ to a line $\ell$ and then on to point $B$ (both $A$ and $B$ on the same side of $\ell$ ), reflect one point across the line.; Points, Lines & Planes › Going Deeper: Advanced Ideas
Right triangles in 3D: the space diagonal: A rectangular box with length $\ell$ , width $w$ , and height $h$ has a space diagonal $d$ running corner to opposite corner.; Right Triangles & Trig › Going Deeper: Advanced Right-Triangle & Trig Ideas
Right triangles: a² + b² = c²: In a right triangle, the two short sides (legs) and the longest side (hypotenuse) satisfy the Pythagorean theorem: $a^2 + b^2 = c^2$ .; Core Ideas in Plain Terms › Triangles and the Pythagorean Theorem
Rigid motions (isometries): These preserve size and shape (image $\cong$ pre-image):; Course Review › Transformations & Symmetry
Rigid Motions vs. Dilations: Transformations › Congruence vs. Similarity
Rotations About Any Center & General Lines: Rotation about a center $(a,b)$ : shift the center to the origin, rotate, then shift back.; Transformations › Going Deeper: Advanced Transformations
Rotations and dilations: Rotations about the origin (counterclockwise):; Course Review › Transformations & Symmetry
Same shape, same size: Two figures are congruent ( $\cong$ ) if they have exactly the same shape and size — one could be slid, turned, or flipped onto the other perfectly.; Triangles & Congruence › Congruent Figures and Corresponding Parts
Same shape, scaled size: Two figures are similar if one is a scaled copy of the other — all angles equal, all sides in the same ratio.; Core Ideas in Plain Terms › Similarity, Area, and Volume
Segment Addition Postulate: If point $B$ lies on $\overline{AC}$ (between $A$ and $C$ ), then the two shorter lengths add to the whole: \[ AB + BC = AC.; Points, Lines & Planes › Measuring Segments & the Segment Addition Postulate
Segment addition, midpoint, and distance: Course Review › Points, Lines, Planes & Segments
Segment Bisector: A segment bisector is any line, ray, or segment that passes through the midpoint of a segment, cutting it into two congruent halves.; Points, Lines & Planes › Segment Bisectors & Basic Constructions
Segments and Rays: Points, Lines & Planes › Segments, Rays, and Opposite Rays
Set Up a Proportion, Then Solve: When two triangles are similar, corresponding sides are proportional.; Similarity › Finding Missing Sides with Similar Triangles
Set up an equation: When an angle relationship gives you a total ( $90^\circ$ , $180^\circ$ , or a bisected angle), write an equation, then solve for the variable.; Angles › Finding Unknown Angles with Algebra
Set up, then solve: When you know one angle and one side and want another side, pick the ratio that uses your two sides (SOH, CAH, or TOA), then solve for the unknown.; Right Triangles & Trig › Using a Trig Ratio to Find a Missing Side
Sharing the total equally: In a regular polygon all $n$ interior angles are equal, so each one is the total divided by $n$ : \[ each interior angle = $\frac{(n-2)\cdot 180^\circ}{n}$ .; Quadrilaterals & Polygons › Each Interior Angle of a Regular Polygon
Similar triangles: Similar figures ( $\sim$ ) have equal corresponding angles and proportional corresponding sides.; Course Review › Similarity
Sine, Cosine, Tangent: Pick one of the two acute angles and call it $\theta$ .; Right Triangles & Trig › The Three Trig Ratios: SOH-CAH-TOA
Slopes of parallel and perpendicular lines: For slope $m$ (recall $m = \dfrac{y_2-y_1}{x_2-x_1}$ , “rise over run”):; Course Review › Parallel & Perpendicular Lines
Special right triangles: Course Review › Right Triangles & Trigonometry
Surface area and volume of solids: Let $B$ = area of the base, $P$ = base perimeter, $h$ = height, $\ell$ = slant height.; Course Review › Perimeter, Area & Volume
Symmetry: Course Review › Transformations & Symmetry
Symmetry Groups & Order of Rotational Symmetry: The set of all rigid motions that map a figure onto itself forms its symmetry group.; Transformations › Going Deeper: Advanced Transformations
Tangent $\perp$ radius: A tangent touches a circle at exactly one point.; Circles › Tangent Lines
The 30-60-90 Triangle: The sides are always in the ratio \[ short leg = x, long leg = x $\sqrt{3}$ , hypotenuse = 2x.; Right Triangles & Trig › Special Right Triangle: 30-60-90
The 45-45-90 Triangle: This is an isosceles right triangle: the two legs are equal, and each base angle is $45^\circ$ .; Right Triangles & Trig › Special Right Triangle: 45-45-90
The altitude on the hypotenuse: geometric means: Drop the altitude from the right angle to the hypotenuse.; Right Triangles & Trig › Going Deeper: Advanced Right-Triangle & Trig Ideas
The Altitude-on-Hypotenuse Geometric-Mean Relations: Drop the altitude from the right angle of a right triangle to the hypotenuse.; Similarity › Going Deeper: Advanced Similarity
The Angle Bisector Theorem: Draw the bisector of one angle of a triangle down to the opposite side.; Triangles & Congruence › Going Deeper: Advanced Triangle Ideas
The Angle Bisector Theorem — via Similarity: Claim.; Similarity › Going Deeper: Advanced Similarity
The angles always add to $180^\circ$: In any triangle, the three interior angles add up to exactly $180^\circ$ .; Triangles & Congruence › The Triangle Angle-Sum Theorem
The apothem and the area of a regular polygon: The apothem $a$ of a regular polygon is the perpendicular distance from the center to the middle of any side.; Quadrilaterals & Polygons › Going Deeper: Advanced Polygon Ideas
The auxiliary parallel line trick (zigzag chases): When a path bends between two parallel lines, there is no single transversal to use.; Parallel & Perpendicular Lines › Going Deeper: Advanced Ideas
The converse statements: Two lines cut by a transversal are parallel if any one of these is true:; Parallel & Perpendicular Lines › Proving Lines Are Parallel (the Converses)
The Exterior Angle Theorem: Extend one side of a triangle to make an exterior angle.; Angles › Going Deeper: Advanced Angle Ideas
The exterior angles always total $360^\circ$: At each vertex, extend one side; the angle between that extension and the next side is an exterior angle.; Quadrilaterals & Polygons › Sum of the Exterior Angles
The foot of a perpendicular: The foot of the perpendicular from a point $P$ to a line $\ell$ is the single point $F$ on $\ell$ where the perpendicular dropped from $P$ lands.; Parallel & Perpendicular Lines › Going Deeper: Advanced Ideas
The four centers of a triangle: Every triangle has four famous “centers,” each built from a different set of special lines.; Triangles & Congruence › Going Deeper: Advanced Triangle Ideas
The full triangle-inequality range: For three sides $a$ , $b$ , $c$ all three sums must beat the remaining side: \[ a+b>c, b+c>a, a+c>b.; Triangles & Congruence › Going Deeper: Advanced Triangle Ideas
The Idea of a Locus: A locus is the set of all points that satisfy a given condition, and nothing else.; Points, Lines & Planes › Going Deeper: Advanced Ideas
The Inscribed Angle Theorem: An inscribed angle has its vertex on the circle, and its two sides are chords.; Circles › Inscribed Angles
The interior-angle-sum formula: Every polygon can be split into triangles by drawing all diagonals from one vertex.; Quadrilaterals & Polygons › Sum of the Interior Angles
The inverse buttons: When you know two sides but want the angle, use the inverse functions $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ (often the “ $2$ nd” + sin/cos/tan keys): \[ θ = ^-1 (opphyp), θ = ^-1 (…; Right Triangles & Trig › Using Inverse Trig to Find a Missing Angle
The Language of Comparison: Similarity › Ratios, Proportions, and Scale Factor
The medial triangle and the Midsegment Theorem: Join the midpoints of the three sides of a triangle and you get a smaller triangle inside, called the medial triangle.; Triangles & Congruence › Going Deeper: Advanced Triangle Ideas
The Midsegment: A midsegment of a triangle joins the midpoints of two sides.; Similarity › The Triangle Midsegment Theorem
The parallel-line angle rules: If two parallel lines are cut by a transversal, then:; Parallel & Perpendicular Lines › Angle Relationships When the Lines Are Parallel
The Power of a Point: If two lines through a point $P$ each meet the circle, the products of the two distances along each line are equal.; Circles › Going Deeper: Advanced Circle Ideas
The Pythagorean Theorem: In any right triangle, the two shorter sides are the legs ( $a$ and $b$ ) and the longest side — always across from the right angle — is the hypotenuse ( $c$ ).; Right Triangles & Trig › The Pythagorean Theorem
The Section (Partition) Formula: The midpoint splits a segment in the ratio $1:1$ .; Points, Lines & Planes › Going Deeper: Advanced Ideas
The Side-Splitter Theorem: If a line is drawn parallel to one side of a triangle and crosses the other two sides, it divides those two sides proportionally.; Similarity › Side-Splitter & Proportional Segments
The slope rules: Parallel lines have equal slopes.; Parallel & Perpendicular Lines › Slopes of Parallel and Perpendicular Lines
The special quadrilaterals: Course Review › Quadrilaterals & Polygons
The tangent–chord angle: When a tangent and a chord meet at the point of tangency, the angle they form is half the intercepted arc — the same “half the arc” rule as an inscribed angle: \[ tangent–chord angle = $\tfrac{1}{2}$ (intercepted arc).; Circles › Going Deeper: Advanced Circle Ideas
The Three Building Blocks: In geometry we start with three ideas so basic that we do not even define them.; Points, Lines & Planes › Undefined Terms: Point, Line, Plane
The three relationships: Parallel lines lie in the same flat surface (plane) and never meet, no matter how far you stretch them.; Parallel & Perpendicular Lines › Kinds of Lines: Parallel, Perpendicular, and Skew
The three-step recipe: Parallel & Perpendicular Lines › Writing Equations of Parallel and Perpendicular Lines
The undefined terms and basic figures: Course Review › Points, Lines, Planes & Segments
Three Shortcuts for Triangles: You do not need to check everything.; Similarity › Triangle Similarity Criteria: AA, SSS $\sim$ , SAS $\sim$
Transformations as Matrices (Preview): Every rotation, reflection, and dilation about the origin can be written as multiplication by a $2\times 2$ matrix acting on the column vector $; Transformations › Going Deeper: Advanced Transformations
Transversal: A transversal is a line that crosses two (or more) other lines.; Parallel & Perpendicular Lines › Kinds of Lines: Parallel, Perpendicular, and Skew
Trapezoid parts: A trapezoid has exactly one pair of parallel sides, called the bases.; Quadrilaterals & Polygons › Trapezoids and Kites
Trigonometric ratios (SOH-CAH-TOA): For an acute angle $\theta$ in a right triangle: \[ θ = opphyp, θ = adjhyp, θ = oppadj .; Course Review › Right Triangles & Trigonometry
Turn a relationship into an equation: Decide whether the two labeled angles are congruent (equal) or supplementary (add to $180^\circ$ ).; Parallel & Perpendicular Lines › Finding Unknown Angles with Algebra
Two chord facts: Circles › Chord Relationships
Two Classic Compass Constructions: Copying a segment $\overline{AB$ : Draw a ray with endpoint $C$ .; Points, Lines & Planes › Segment Bisectors & Basic Constructions
Two counting rules that act as constraints: These two facts are the “budget” every angle problem must respect: \[ along a line: ∑ angles = 180^°, around a point: ∑ angles = 360^° .; Angles › Going Deeper: Advanced Angle Ideas
Two special sums: Angles › Complementary & Supplementary Angles
Two ways to sort a triangle: Every triangle can be described in two ways at once: by its sides and by its angles.; Triangles & Congruence › Classifying Triangles
Types of angles: Measured in degrees ( $^\circ$ ): acute $<90^\circ$ , right $=90^\circ$ , obtuse between $90^\circ$ and $180^\circ$ , straight $=180^\circ$ .; Course Review › Angles & Angle Relationships
Varignon's theorem: the midpoint quadrilateral: Take any quadrilateral and mark the midpoint of each of its four sides.; Quadrilaterals & Polygons › Going Deeper: Advanced Polygon Ideas
Vertical angles: When two straight lines cross, they form two pairs of vertical angles — the angles “across” from each other.; Angles › Linear Pairs & Vertical Angles
What “Similar” Means: Two polygons are similar ( $\sim$ ) when they have the same shape but not necessarily the same size.; Similarity › Similar Polygons
What a proof looks like: A two-column proof lists Statements on the left and the Reason for each on the right.; Triangles & Congruence › A First Two-Column Proof
What Happens When You Resize a Shape: If you multiply every length of a figure by the same factor $k$ :; Area & Volume › Scaling Dimensions & Real-World Problems
What is a bisector?: A ray that bisects an angle divides it into two congruent (equal) angles.; Angles › Angle Bisectors
What is a polygon?: A polygon is a closed, flat shape made of straight line segments.; Quadrilaterals & Polygons › Polygon Vocabulary
What is an angle?: An angle is formed by two rays that share a common endpoint.; Angles › Naming Angles and Their Parts
What makes a parallelogram?: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.; Quadrilaterals & Polygons › Properties of Parallelograms
Where the 30-60-90 ratio comes from: Take an equilateral triangle with every side $2x$ and every angle $60^\circ$ .; Right Triangles & Trig › Going Deeper: Advanced Right-Triangle & Trig Ideas
Where the 45-45-90 ratio comes from: The ratio is not memorized magic — it follows from the Pythagorean Theorem.; Right Triangles & Trig › Going Deeper: Advanced Right-Triangle & Trig Ideas
Who is who: Every square is a rectangle and a rhombus; every rectangle and every rhombus is a parallelogram; every parallelogram is a quadrilateral.; Quadrilaterals & Polygons › The Quadrilateral Family Tree
Whole-number right triangles: A Pythagorean triple is a set of three whole numbers that fit $a^2+b^2=c^2$ exactly.; Right Triangles & Trig › Pythagorean Triples
Why the perpendicular from the center bisects a chord: Section on chords stated it; here is why.; Circles › Going Deeper: Advanced Circle Ideas
Why the Side-Splitter Theorem Is True (A Proof): Claim.; Similarity › Going Deeper: Advanced Similarity
Why the transversal theorems are true: The whole system rests on one assumed fact, the Corresponding Angles Postulate: if two parallel lines are cut by a transversal, corresponding angles are congruent.; Parallel & Perpendicular Lines › Going Deeper: Advanced Ideas