What you'll learn

• Classifying Polygons and Special QuadrilateralsPolygons are sorted into categories based on properties such as number of sides, side lengths, and angle measures to instantly know the characteristics of each shape.
• Area of PolygonsAny polygon's area can be computed using formulas for basic shapes like triangles, parallelograms, and trapezoids, or by decomposing irregular figures into simpler components.
• Pythagorean Theorem and ApplicationsFor any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs.
• Structure of Geometric ProofA logical argument that uses definitions, postulates, and previously proven theorems to establish that a new geometric statement must be true.
• Coordinate Proofs of Geometric TheoremsGeometric statements are proved by placing figures on the coordinate plane and using algebra, slope, distance formulas, and midpoint formulas to verify the required relationships.
• Dilations and Scale FactorA dilation is a transformation that enlarges or shrinks a figure by a scale factor from a fixed center point, preserving shape and angle measure but changing size.
• Modeling with Geometric ShapesRepresenting real-world objects using idealized geometric shapes allows you to apply mathematical formulas and reasoning to solve practical problems.
• Central and Inscribed AnglesAn inscribed angle is half the measure of the central angle that intercepts the same arc.
• Triangle Congruence CriteriaMinimal sets of corresponding parts—SSS, SAS, ASA, AAS, and HL—that guarantee two triangles are identical in shape and size without checking all sides and angles.
• Distance and Midpoint FormulasThe distance formula calculates the length of a segment between two points using their coordinates and the Pythagorean theorem.
• Cavalieri's Principle and Volume ReasoningTwo solids have equal volume if they are stacked between two parallel planes and every horizontal slice through both solids at the same height has equal cross-sectional area.
• Proving Theorems About TrianglesFoundational theorems such as the angle sum property and isosceles base angles, proved by drawing auxiliary lines and applying congruence criteria to establish facts that hold for all triangles of a given type.
• Parallel Lines and TransversalsWhen a transversal crosses two parallel lines, it creates eight angles at two intersection points that fall into predictable relationships, allowing you to find any angle measure once you know one.
• Surface Area and Volume of Prisms and CylindersPrisms are three-dimensional solids with two parallel, congruent polygonal bases connected by rectangular faces, while cylinders have circular bases instead, and their surface areas can be calculated by unfolding them into nets showing all faces laid flat.
• Segments, Rays, and Angle BasicsLine segments are finite parts of lines with two endpoints, while rays extend infinitely from a single endpoint in one direction.
• Right Triangle Trigonometric RatiosSine, cosine, and tangent are ratios of specific pairs of sides in a right triangle that depend on a chosen acute angle.
• Rigid Motions: Translations, Reflections, RotationsTransformations that move figures without changing distances or angle measures are the foundation for defining congruence between shapes.
• Surface Area and Volume of Pyramids and ConesPyramids and cones are three-dimensional solids where triangular or curved lateral faces taper to a single apex point, each containing exactly one-third the volume of a prism or cylinder sharing the same base and height.
• Proving Theorems About ParallelogramsA set of properties of quadrilaterals with both pairs of opposite sides parallel, provable through triangle congruence using diagonals and alternate interior angles, with converses that identify sufficient conditions for a quadrilateral to be a parallelogram.
• Proving Theorems Using SimilarityEstablishing triangle similarity through angle and side relationships allows you to use proportional side lengths as justification for geometric theorems.
• Congruence Defined by Rigid MotionsTwo figures are congruent if and only if a sequence of rigid motions—translations, rotations, and reflections—can map one figure exactly onto the other.
• Points, Lines, and PlanesThe three undefined foundational terms of geometry—a point as a locationless position, a line as a straight one-dimensional figure extending infinitely, and a plane as a flat two-dimensional surface—from which all other geometric objects are defined.
• Triangle Similarity CriteriaMinimal conditions—AA, SAS, and SSS—that guarantee two triangles have the same shape with proportional corresponding sides, without verifying all angles and side lengths.
• Symmetry of FiguresA figure has symmetry when a rigid motion other than the identity—such as reflection across a line or rotation about a point—maps the figure exactly onto itself, with line symmetry occurring when reflection across a line preserves the figure and rotational symmetry occurring when rotation about a point does so.
• Similarity Defined by TransformationsTwo figures are similar if a sequence of rigid motions and dilations maps one exactly onto the other.
• Compass-and-Straightedge ConstructionsExact geometric figures are drawn using only a compass and straightedge, with precision guaranteed by the logical steps of the construction rather than freehand estimation.
• Solving Right TrianglesFinding all three side lengths and three angle measures of a right triangle using the Pythagorean theorem and trigonometric ratios.
• Arcs and Chords RelationshipsIn a circle, two chords are congruent if and only if their corresponding arcs are congruent, a relationship that follows from the triangle congruence of the radii and chord segments.
• Inscribed and Circumscribed PolygonsA polygon is inscribed in a circle when all its vertices lie on the circle, and circumscribed about a circle when all its sides are tangent to the circle.
• Inductive and Deductive ReasoningInductive reasoning forms a general conjecture by observing patterns in specific examples, while deductive reasoning applies established rules to reach a conclusion that must logically follow.
• Slope Criteria for Parallel and Perpendicular LinesTwo distinct non-vertical lines are parallel if and only if they have the same slope, and two lines are perpendicular if and only if their slopes are negative reciprocals.
• Tangent Lines to CirclesA tangent line is a line that touches a circle at exactly one point and is perpendicular to the radius drawn to that point of tangency.
• Equation of a CircleThe standard form equation of a circle encodes the center and radius by expressing all points equidistant from a fixed center point.
• Special Angle Pair RelationshipsPairs of angles can be classified by how their measures relate, such as complementary angles summing to 90 degrees or supplementary angles summing to 180 degrees.
• Logical Statements and Conditional ReasoningA conditional statement uses if-then form to express a logical relationship between a hypothesis and conclusion, forming the foundation for geometric reasoning and proof.
• Surface Area and Volume of SpheresA sphere is the set of all points in three-dimensional space at a fixed distance (radius) from a center point, with volume equal to four-thirds pi times the radius cubed and surface area equal to four times pi times the radius squared.
• Arc Length and Sector AreaThe measure of a central angle determines what fraction of a circle's circumference forms an arc and what fraction of the circle's area forms a sector, following a single proportional relationship.
• Density and Geometric ModelingDensity measures how much of a quantity is packed into a given amount of space or area by dividing the quantity by the area or volume.
• Area and Perimeter Using CoordinatesThe perimeter of a polygon with known vertices is the sum of distances between consecutive vertices, and area can be found either by decomposing the polygon into simpler shapes or by using coordinate-based formulas.
• Cross-Sections and Solids of RevolutionA cross-section is the two-dimensional shape formed when a plane slices through a three-dimensional solid, with its shape depending on both the solid and the angle and position of the cut.
• Partitioning a Segment in a Given RatioA method for finding the coordinates of a point that divides a segment into a specified ratio, which generalizes the midpoint formula and applies proportional reasoning to geometric placement.