Geometry

About this course

This is a dedicated Geometry course, and proof is at the center of it. The year opens with points, lines, and planes, the ruler and protractor postulates, angle pairs, segment and angle bisectors, and the coordinate plane, then gives a full unit to logic and proof: inductive and deductive reasoning, conditional statements with their converse, inverse, and contrapositive, biconditionals, the laws of detachment and syllogism, the properties of equality used as reasons, and how to write a two-column, paragraph, or flow proof. From there come parallel lines and transversals, triangle congruence by SSS, SAS, ASA, and AAS, isosceles and equilateral triangles, the triangle inequality and midsegment theorems, similarity by AA, SAS, and SSS, the 45-45-90 and 30-60-90 special triangles, the Pythagorean theorem and its converse, quadrilaterals and polygon angle formulas, surface area and volume of prisms, pyramids, cones, and spheres, and a full unit on circles covering arcs, central and inscribed angles, sectors, chords, secants, and tangents.

The second half goes further than a standard geometry year. It covers transformations (reflections, rotations, translations, compositions, tessellations, and an introduction to fractals); right triangle trigonometry extended through inverse functions, the law of sines, the law of cosines, vectors and the dot product, and navigation by bearing; coordinate geometry with coordinate proofs, partitioning a segment, and locus; compass-and-straightedge construction, including the classical Greek problems and the golden ratio; analytic geometry with parabolas, ellipses, and hyperbolas identified by completing the square; geometric probability with Monte Carlo methods and expected value; three-dimensional geometry with polyhedra, Euler's formula, cross-sections, and nets; and a closing modeling unit that applies geometry to architecture, navigation, computer graphics, and conservation. The course is 160 lessons in 15 units of 10 to 12, at about 55 minutes each, which is a full year with a little slack in it since the daily-scheduled courses run 180. A lesson opens with prayer, states the definitions, postulates, or theorems for the day, works two to four examples with each step justified, adds an application and a short worldview note tied to a KJV verse, and ends with a ten-question multiple-choice quiz that grades itself. The writing is dense and compact, so the worked examples carry much of the teaching. Most units close with a review lesson, and the last unit reviews the whole year.

Materials are pencil, graph paper, a compass, a ruler, a protractor, and a Bible. The compass and protractor are genuinely used, since one full unit is straightedge-and-compass construction. Your student should finish Algebra I before starting. Geometry here leans on algebra constantly — solving equations for a missing side, slope and the equations of lines, the distance and midpoint formulas, simplifying radicals, and completing the square to identify a conic are all used without being retaught — and the trigonometry, vector, and conics units push past what a typical geometry course includes, so a student who is shaky on algebraic manipulation will struggle in the later units. Freedom Homeschool also offers a general Math course at tenth grade, and the two overlap. That course is chiefly advanced algebra and precalculus, with complex numbers, sequences and series, polynomial and rational functions, matrices, and logarithms, and geometry appears there as one strand among several. This course is the opposite balance: a full year of plane geometry with formal proof taught as a skill in its own right, which the general course touches but does not develop at this length. On a standard college-prep path Geometry usually comes after Algebra I and before advanced algebra, so for most families this is the tenth-grade math course rather than something added to it.

What the course covers

• Points, lines, planes, angle pairs, and the coordinate plane
• Logic, conditional statements, and two-column, paragraph, and flow proofs
• Parallel lines, transversals, and triangle congruence by SSS, SAS, ASA, and AAS
• Similarity, special right triangles, and the Pythagorean theorem
• Quadrilaterals, polygon angle formulas, area, surface area, and volume
• Circles: arcs, inscribed angles, sectors, chords, secants, and tangents
• Transformations, symmetry, tessellations, and dilations
• Right triangle trigonometry, the law of sines, the law of cosines, and vectors
• Compass-and-straightedge constructions and the classical Greek problems
• Conic sections, geometric probability, and three-dimensional geometry

What your student will be able to do

• Write a two-column proof with a justified reason for every statement
• Prove two triangles congruent or similar and use the result to find missing measures
• Apply circle theorems to find arc measures, angles, and segment lengths
• Solve for missing sides and angles with trigonometric ratios and the laws of sines and cosines
• Construct bisectors, perpendiculars, parallels, and regular polygons with compass and straightedge
• Find the surface area and volume of prisms, pyramids, cones, spheres, and composite solids

Scope and sequence

All 15 units of Geometry for Grade 10, in the order they are taught.

Every lesson, by title

All 160 lesson titles are listed here — nothing is withheld until you download. Open a unit to read its contents.

One whole lesson, exactly as it comes

This is a real lesson taken from the middle of the course — the teaching text, the practice, the homework, and the first three quiz questions with their answers. It is not a marketing excerpt.

What you will need

Drawn from the materials named across all 160 lessons of this course. Most families already own most of it.