Online Geometry Course for Grades 8 and 9: Proofs, Triangles, Circles and Trigonometry

Points, lines and planes, segment and angle measure with the Midpoint and Distance Formulas, constructions, angle pairs, polygons, and perimeter and area.

• Elements of Geometry and Segment AdditionUndefined terms (point, line, plane), postulates, rays and opposite rays, collinear and coplanar points; then distance and length, the Segment Addition Postulate, congruent segments, midpoints and segment bisectors.
• Distance, Midpoint, Angles and Angle AdditionThe Midpoint Formula and Distance Formula in the coordinate plane, a construction of a segment's midpoint, then naming, measuring and classifying angles, congruent angles, the Angle Addition Postulate and angle bisectors.
• Angle Pairs and PolygonsAdjacent angles, linear pairs, vertical angles, complementary and supplementary angles; then polygons (convex or concave, equilateral, equiangular, regular), the angle sum of a polygon, and perimeter and area of basic shapes.

Properties of equality and two-column proofs, angles formed by a transversal and the converses that prove lines parallel, then slope and the equations of lines.

• Proofs by Reasoning and CongruencyProperties of Equality (reflexive, symmetric, transitive) and the Distributive Property used as reasons in proofs, then the theorems on congruent segments and angles, right angles, congruent supplements and congruent complements.
• Transversals and Angle RelationshipsParallel, skew and perpendicular lines and parallel planes, transversals and the angle pairs they form, the Corresponding Angles Postulate and the theorems for two parallel lines cut by a transversal.
• Angle Converses and Equations of LinesConverses of the angle theorems for proving lines parallel; then slope, slopes of parallel and perpendicular lines, and slope-intercept, point-slope and standard form of a line, with algebra applications.

Triangle classification and angle sums, congruence (SSS, SAS, HL, ASA, AAS) and CPCTC, isosceles triangles, bisectors and the triangle centers, midsegments, the Pythagorean Theorem and the triangle inequalities.

• Triangles and CongruencyClassifying triangles by sides and by angles, the Triangle Sum Theorem and Exterior Angle Theorem; then congruent figures, corresponding parts, congruence statements, the Third Angles Theorem and properties of congruent triangles.
• CPCTC and Isosceles TrianglesUsing CPCTC after proving triangles congruent, in two-column proofs, coordinate proofs and real-world problems; then the parts of an isosceles triangle, the Base Angles Theorem, its converse and the corollaries.
• Angle and Perpendicular BisectorsEquidistant points and locus, the Perpendicular Bisector Theorem and the Angle Bisector Theorem with their converses, then concurrent lines, the circumcenter and incenter of a triangle and its circumscribed and inscribed circles.
• Median, Altitude, Centroid, Orthocenter, IncenterMedians and the centroid, altitudes and the orthocenter, a recap of the special segments and centers of a triangle, the median to the hypotenuse, and mixed problems on the circumcenter and incenter.
• Midsegment, Pythagorean Theorem, Triangle InequalitiesThe midsegment of a triangle and the Midsegment Theorem; the Pythagorean Theorem with a proof, Pythagorean triples and its converse; and the two triangle inequality theorems relating sides to opposite angles.

Ratios, proportions and scale, similar polygons, the AA, SSS and SAS similarity tests, proportionality theorems, and a review week.

• Ratios, Proportions, Similar PolygonsRatios, proportions and the Cross Product Property, scale drawings, then similar polygons: scale factor, similarity ratio, identifying similar polygons and similar triangles formed by parallel sides.
• Triangle Similarity, ProportionalityThe Perimeter Theorem for similar polygons, the AA Similarity Postulate, the Triangle Proportionality Theorem and its converse, three parallel lines, the angle bisector proportion, and SSS and SAS similarity. A challenge covers the golden ratio.

Special right triangles, polygon angle sums, the tangent ratio, sine and cosine, geometric mean, inverse trig, and the Law of Sines and Law of Cosines.

• Angles in Polygons and Tangent RatioPolygon definitions, the interior-angle sum (n − 2)·180°, one angle of a regular polygon and the exterior-angle sum of 360°; then the tangent ratio, angles of elevation and depression, and tan⁻¹ to find an angle.
• Trigonometric Ratios and Geometric MeanNaming the sides of a right triangle, sine, cosine and tangent (SOH-CAH-TOA) for sides, trig values, slope and area, and inverse trig for angles; then the geometric mean with the Altitude and Leg Theorems.
• Law of Sines and CosinesA recap of right-triangle trig and inverse trig, then the Law of Sines and Law of Cosines for any triangle, choosing between them, classifying a triangle by its sides, and the identity sin²α + cos²α = 1 with special-angle values.

Parallelograms, rhombuses, rectangles, squares, trapezoids and kites with their theorems and area formulas, then translations, reflections, rotations and dilations.

• Quadrilaterals (Parallelograms)Quadrilaterals and parallelograms: opposite sides and angles congruent, consecutive angles supplementary, diagonals bisecting each other; then five ways to prove a quadrilateral is a parallelogram, with algebra and coordinates.
• Special ParallelogramsRhombus, rectangle and square: their defining properties and theorems, a Venn diagram of the parallelogram family, the diagonal theorems, and classifying a quadrilateral. Includes a review of polygon angle sums.
• Trapezoids and KitesTrapezoids (bases, legs, base angles, isosceles trapezoids, the midsegment theorem) and kites; classifying all the special quadrilaterals; and deriving area formulas for the parallelogram, rhombus and trapezoid.
• TransformationsTranslations, reflections, rotations and dilations with their coordinate rules, identifying a transformation from a preimage and image, and real-world uses. The week closes with a review of the Law of Sines and Law of Cosines.

Parts of a circle, arcs and circumference, tangents, inscribed angles, angles and segments formed by chords, secants and tangents, and the equation of a circle.

• CirclesParts of a circle, circumference C = πd = 2πr, tangents and secants with the Tangent Line Theorem, central angles and arcs (minor, major, semicircle), arc length, and congruent circles and arcs.
• Circles with Other Geometric FiguresInscribed angles and their intercepted arcs, inscribed polygons and quadrilaterals, tangent-chord angles, angles formed inside and outside a circle, and concentric and inscribed circles with tangent lengths.
• Circles and PropertiesThe Segments of Chords, Segments of Secants, and Segments of a Secant and a Tangent theorems, with proofs and vocabulary, followed by mixed problems on arcs, angles, chords and tangent-radius right triangles.
• Equation of a CircleThe equation of a circle centered at the origin, x² + y² = r², and the standard form (x − h)² + (y − k)² = r²; writing it from a graph, a diameter or a point, completing the square, and tangent and secant lines.

Areas of polygons and circles, sectors, surface area and volume of solids, three-dimensional coordinates, and a mixed review of the whole course.

• Areas of PolygonsAreas of squares, rectangles, triangles and parallelograms, then trapezoids, rhombuses and kites, areas from coordinates, perimeters and areas of similar polygons, and a summary of the area formulas.
• Areas and Circumference of Circles and PolygonsCircumference and area of a circle, arc length and the area of a sector, shaded regions and composite figures, and the area of a regular polygon using its apothem and central angle.
• Polyhedra, Nets, Surface AreaPolyhedra, prisms and the regular (Platonic) solids, cross sections and nets, then surface area of prisms, pyramids, cylinders, cones and spheres. A challenge covers Euler's formula.
• Volume of Basic SolidsVolume of prisms, cylinders, pyramids, cones and spheres, oblique solids and Cavalieri's Principle, truncated cones, volumes of similar solids (ratio a³ : b³), and Euler's formula.
• Geometry in SpaceOrdered triples, the Distance and Midpoint Formulas in space, solids cut by a plane parallel to the base, and surface area and volume of composite solids such as a castle, an hourglass and a capsule tank.