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High School Geometry: Proof, Shape, and Measurement That Make Sense
This page covers the full high school Geometry course in the Common Core traditional pathway: transformations and congruence, similarity and right-triangle trigonometry, formal proof, circles, coordinate geometry including conic sections, volume of solids, geometric modeling, and conditional probability with expected value. It assumes you've finished Algebra I — you can solve linear equations, work with exponents and square roots, use the Pythagorean theorem, and graph lines.
A lesson here is a conversation, not a lecture. The tutor asks you to notice something about a figure, predict what a transformation does, or try the next step of a proof yourself, and responds to what you actually say — including when you're partly right. Over time it checks that ideas you learned weeks ago (like why alternate interior angles are congruent) still hold up when they resurface in a new proof about parallelograms or circles.
Who it is for
- High school students taking Geometry who want to work through proofs and problems by talking them out
- Students who found definitions and theorems hard to remember and want to rebuild them from reasoning instead
- Parents looking for steady, topic-by-topic coverage matching a standard Geometry course sequence
- Teachers wanting a supplementary resource that mirrors the Common Core Appendix A traditional pathway
- Students preparing for a unit test on congruence, similarity, circles, or probability who want targeted review
What you will be able to do
- Write and follow formal proofs about lines, angles, triangles, and parallelograms using rigid motions and similarity
- Use congruence and similarity criteria (SAS, ASA, SSS, AA) to solve for unknown lengths and angles
- Apply right-triangle trigonometry, the Law of Sines, and the Law of Cosines to find missing sides and angles in any triangle
- Derive and use circle relationships: inscribed angles, tangents, arc length, and the equation of a circle
- Set up coordinate proofs and find equations of circles, parabolas, ellipses, and hyperbolas from geometric definitions
- Compute volumes of prisms, cylinders, cones, pyramids, and spheres, and use conditional probability and expected value to analyze decisions
A taste of a lesson on Triangle congruence: ASA, SAS, SSS
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 40 topics in Geometry
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Undefined terms and precise definitions
Point, line, plane, and distance along a line or around a circular arc are undefined notions. Build precise definitions from them of angle, circle, perpendicular lines, parallel lines, and line segment. Students should be able to state a definition precisely and tell whether a loose description (like 'a corner') meets it.
Transformations as functions of the plane
Rotations, reflections, and translations, and dilations, described as functions that map input points to output points. Define each in terms of angles, circles, perpendicular lines, parallel lines, and segments (e.g., a reflection across line l maps each point to the point on the perpendicular through it at equal distance on the other side). Students draw transformed figures on graph paper or with software and specify a sequence of transformations carrying one figure onto another. Distinguish transformations that preserve distance and angle from those that do not.
Builds on Undefined terms and precise definitions
Rigid motions and congruence
Rigid motions (compositions of rotations, reflections, translations) preserve distance and angle measure. Two figures are congruent when one is carried onto the other by a sequence of rigid motions. Use this to decide whether given figures are congruent and to predict the effect of a rigid motion on a figure.
Builds on Transformations as functions of the plane
Symmetry of rectangles, parallelograms, trapezoids, and regular polygons
For a given rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry the figure onto itself. Symmetries are rigid motions that map the figure onto itself. Count rotational and reflectional symmetries of regular polygons and identify which quadrilaterals have which symmetries.
Builds on Transformations as functions of the plane, Rigid motions and congruence
Triangle congruence: ASA, SAS, SSS
Show that two triangles are congruent if and only if corresponding sides and angles are congruent (CPCTC). Explain how the SAS, ASA, and SSS criteria follow from the rigid-motion definition: a rigid motion can carry one vertex, then one side, then the third vertex into place. Students use the criteria to decide congruence and to find unknown parts.
Builds on Undefined terms and precise definitions, Rigid motions and congruence
Compass-and-straightedge constructions
Construct with a variety of tools: copy a segment, copy an angle, bisect a segment, bisect an angle, construct perpendicular lines (including through a point on or off a line), and construct parallel lines through a point. Also construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Students justify why each construction works using congruence or definitions.
Builds on Undefined terms and precise definitions, Triangle congruence: ASA, SAS, SSS
Proofs about lines and angles
Prove theorems about lines and angles: vertical angles are congruent, angles formed by parallel lines cut by a transversal (corresponding, alternate interior, alternate exterior, same-side interior) are congruent or supplementary, and points on a perpendicular bisector are equidistant from segment endpoints. Students write two-column and paragraph proofs and understand the logical structure of a proof.
Builds on Undefined terms and precise definitions, Triangle congruence: ASA, SAS, SSS
Proofs about triangles
Prove the triangle sum theorem (interior angles sum to 180 degrees), the exterior angle theorem, the isosceles triangle theorem and its converse, and that the midsegment joining midpoints of two sides is parallel to the third side and half its length. Prove that the medians of a triangle are concurrent (meet at a centroid that divides each median 2:1 from the vertex). Use these to solve problems about triangles.
Builds on Proofs about lines and angles
Interior and exterior angle sums of polygons
Prove that the interior angles of a convex n-gon sum to (n-2)*180 degrees by triangulating from one vertex, and that the exterior angles (one at each vertex) sum to 360 degrees. Apply these to find the measure of each interior angle of a regular polygon and the number of sides from a given angle measure.
Builds on Proofs about triangles
Proofs about parallelograms
Prove that opposite sides and opposite angles of a parallelogram are congruent, that diagonals bisect each other, and the converses that establish a quadrilateral as a parallelogram. Also prove properties of rectangles, rhombi, and squares follow from parallelogram properties and the diagonal properties.
Builds on Proofs about triangles, Interior and exterior angle sums of polygons
Triangle inequalities and relationships in triangles
Use the rule that larger sides lie opposite larger angles, the triangle inequality, and the hinge theorem to compare lengths and angles in triangles. Solve problems about which side lengths are possible.
Builds on Proofs about triangles
Dilations and similarity transformations
Verify experimentally that a dilation with center and scale factor takes a line not through the center to a parallel line and leaves a line through the center unchanged, and that a segment is multiplied in length by the scale factor. Define similarity as a sequence of similarity transformations (rigid motions plus dilations), decide if two figures are similar, and explain that similar triangles have all corresponding angles equal and sides proportional.
Builds on Transformations as functions of the plane, Rigid motions and congruence
AA similarity criterion
Use properties of similarity transformations to establish that two triangles with two pairs of congruent angles are similar. Use AA to find unknown lengths in similar triangles and to set up proportions.
Builds on Proofs about triangles, Dilations and similarity transformations
Similarity theorems and applications of triangle similarity
Prove theorems about triangles using similarity, including the side-splitting theorem (a line parallel to one side divides the other two sides proportionally) and its converse, and the triangle midsegment via similarity. Use SSS and SAS similarity criteria together with AA to solve problems, including indirect measurement such as finding heights and distances with shadows or mirrors.
Builds on Triangle congruence: ASA, SAS, SSS, AA similarity criterion
Pythagorean theorem and its proofs
State the Pythagorean theorem and the converse; prove it using similarity of the three triangles formed by an altitude to the hypotenuse; use it to find missing side lengths and to decide whether a triangle is right, acute, or obtuse. Apply it to distances in simple plane figures.
Builds on Similarity theorems and applications of triangle similarity
Trigonometric ratios of acute angles
Recognize that side ratios in right triangles depend only on the acute angles (by similarity). Define sine, cosine, and tangent of an acute angle as opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent. Explain and use that sine and cosine of complementary angles are equal to each other's values (sin(θ)=cos(90°−θ)).
Builds on Pythagorean theorem and its proofs
Solving right triangles in applied problems
Use trigonometric ratios and the Pythagorean theorem to find all missing sides and angles of right triangles in applied contexts such as angles of elevation and depression, heights, and distances. Include setting up the diagram and checking reasonableness of answers.
Builds on Trigonometric ratios of acute angles
Area of a triangle using sine
Derive A = (1/2)ab sin(C) by drawing an altitude from a vertex perpendicular to the opposite side, using right triangle trig. Use the formula to find areas when two sides and the included angle are known.
Builds on Solving right triangles in applied problems
Laws of Sines and Cosines
Prove the Law of Sines and the Law of Cosines (by dropping an altitude or using coordinates, and recognizing that the Law of Cosines reduces to the Pythagorean theorem when the angle is 90 degrees). Use them to find unknown sides and angles in non-right triangles, including SSA ambiguity and applications such as surveying and resultant forces.
Builds on Triangle inequalities and relationships in triangles, Area of a triangle using sine
Circles: similarity, arcs, and radian measure
Prove all circles are similar using dilations. Derive that arc length intercepted by a central angle is proportional to the radius, define radian measure as the constant of proportionality (one radian is the angle whose arc equals the radius), and derive the sector area formula A = (1/2) r^2 theta (theta in radians).
Builds on Compass-and-straightedge constructions, Dilations and similarity transformations
Inscribed angles, radii, and chords
Identify and prove relationships among central angles, inscribed angles, radii, and chords: an inscribed angle is half its intercepted arc; angles inscribed in a semicircle are right; perpendicular from center to a chord bisects the chord; equal chords are equidistant from center.
Builds on Proofs about triangles, Circles: similarity, arcs, and radian measure
Inscribed and circumscribed circles, cyclic quadrilaterals, and tangents
Construct the inscribed circle (incenter, angle bisectors) and circumscribed circle (circumcenter, perpendicular bisectors) of a triangle. Prove that opposite angles of a quadrilateral inscribed in a circle are supplementary. Construct a tangent line from a point outside a circle to the circle (using the Thales circle on the segment from the center to the point) and prove a tangent is perpendicular to the radius at the point of tangency.
Builds on Compass-and-straightedge constructions, Inscribed angles, radii, and chords
Coordinate plane and slope
Plot points in the coordinate plane; define slope as rise over run and compute it from two points; interpret slope and intercept of a line; write equations of lines in slope-intercept and point-slope form. This is the algebraic base for coordinate proofs.
Coordinates: distance, midpoint, and directed segment partitions
Use the distance formula derived from the Pythagorean theorem and the midpoint formula. Find the point on a directed segment from one point to another that partitions it in a given ratio (e.g., the point 2/5 of the way from A to B) using the section formula derived from similar triangles. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.
Builds on Pythagorean theorem and its proofs, Coordinate plane and slope
Slope criteria for parallel and perpendicular lines
Prove that two non-vertical lines are parallel iff they have equal slopes, and perpendicular iff the product of slopes is -1. Use these to find the equation of a line parallel or perpendicular to a given line through a given point, and to solve geometric problems such as identifying right angles in figures.
Builds on Proofs about lines and angles, Coordinate plane and slope
Coordinate proofs of geometric theorems
Place figures in convenient positions on the coordinate plane and prove simple theorems algebraically, such as showing a quadrilateral with four given vertices is a rectangle (using slope and distance), or that a given point lies on a circle. Discuss choosing coordinates to simplify the work.
Builds on Coordinates: distance, midpoint, and directed segment partitions, Slope criteria for parallel and perpendicular lines
Equations of circles
Derive the equation (x-h)^2 + (y-k)^2 = r^2 from the Pythagorean theorem. Complete the square to convert a general quadratic equation of a circle into standard form and read off the center and radius.
Builds on Coordinates: distance, midpoint, and directed segment partitions
Parabolas, ellipses, and hyperbolas from foci
Derive the equation of a parabola from a focus and directrix (points equidistant from both). For ellipses and hyperbolas (+), derive equations from the foci using the fact that the sum (ellipse) or difference (hyperbola) of distances from the two foci is constant.
Builds on Equations of circles
Cross-sections and solids of revolution
Identify the shapes of two-dimensional cross-sections of three-dimensional objects (slicing a cube or cylinder by planes), and identify three-dimensional solids generated by rotating a two-dimensional figure (a rectangle around an axis gives a cylinder, a right triangle around a leg gives a cone, a semicircle gives a sphere).
Builds on Compass-and-straightedge constructions, Pythagorean theorem and its proofs
Informal arguments for circumference and area of a circle
Give informal arguments for the formulas C = 2 pi r and A = pi r^2, for instance by dividing a circle into sectors and rearranging them into an approximate parallelogram, and by the relation between circumference and area through the radius. Use these formulas in problems.
Builds on Circles: similarity, arcs, and radian measure
Volume of prisms and cylinders via Cavalieri-style reasoning
Give an informal argument that the volume of a prism or cylinder is base area times height (stacking congruent cross-sections). Compute volumes of prisms and cylinders and use the formulas in problems.
Builds on Cross-sections and solids of revolution, Informal arguments for circumference and area of a circle
Volume of pyramids, cones, and spheres
Give an informal argument for the volume formulas of a pyramid (one third of base area times height) and a cone (one third of pi r^2 h), using comparison with prisms and cylinders. Give an informal argument using Cavalieri's principle (+) for the volume of a sphere, V = (4/3) pi r^3, and for other solids. Use all volume formulas to solve problems.
Builds on Volume of prisms and cylinders via Cavalieri-style reasoning
Modeling objects and density with geometry
Use geometric shapes, their measures, and properties to describe real objects (modeling a tree trunk or a human torso as a cylinder, a can as a cylinder). Apply density concepts based on area and volume (people per square mile, BTUs per cubic foot), and set up and interpret these models with units.
Builds on Informal arguments for circumference and area of a circle, Volume of pyramids, cones, and spheres
Geometric design problems
Apply geometric methods to solve design problems: design an object or structure to satisfy physical constraints or minimize cost (e.g., a can of a given volume with least material, a box from a sheet with cutouts), and work with ratio-based systems such as typographic grid systems and aspect ratios.
Builds on Similarity theorems and applications of triangle similarity, Modeling objects and density with geometry
Sample spaces, events, and set operations
Describe events as subsets of a sample space using characteristics of outcomes, and as unions, intersections, and complements (or, and, not). Use two-way frequency tables as sample spaces to classify data by two categories.
Builds on Undefined terms and precise definitions
Addition rule and independence
Apply the Addition Rule P(A or B) = P(A) + P(B) - P(A and B) and interpret the result. Understand that A and B are independent when P(A and B) = P(A)P(B), and use this to test independence in examples and two-way tables.
Builds on Sample spaces, events, and set operations
Conditional probability
Define the conditional probability of A given B as P(A and B)/P(B), and as the fraction of B's outcomes that also lie in A. Interpret independence as P(A|B)=P(A). Use two-way tables to approximate conditional probabilities and recognize conditional probability in everyday language (e.g., chance of disease given a positive test vs. chance of positive test given disease).
Builds on Addition rule and independence
General multiplication rule and counting
Apply P(A and B) = P(A)P(B|A) = P(B)P(A|B) in uniform probability models. Use permutations and combinations to count outcomes and compute probabilities of compound events.
Builds on Conditional probability
Random variables and expected value
Define a random variable by assigning numbers to outcomes and graph its probability distribution. Calculate expected value as the mean of the probability distribution, for both theoretical distributions (e.g., sum of two dice) and empirical distributions (e.g., TV sets per household).
Builds on General multiplication rule and counting
Expected payoff and fair decisions
Assign probabilities to payoff values to weigh the possible outcomes of a decision; find expected payoff of a game of chance (e.g., lottery ticket), and compare strategies by expected value (e.g., high vs. low deductible insurance). Use probability to make fair decisions such as drawing lots or random number generation, and analyze decisions with probability such as medical testing or product testing.
Builds on Random variables and expected value
Practice questions in Geometry
A few of the kinds of questions you will work through, with answers.
In triangle ABC, angle A = 54 degrees and angle B = 61 degrees. What is angle C, and what is the measure of the exterior angle at C?
See the answer
Angle C = 65 degrees; exterior angle at C = 115 degrees
Interior angles sum to 180, so C = 180 - 54 - 61 = 65. The exterior angle is supplementary to C, or equivalently equals the sum of the two remote interior angles (54 + 61 = 115).
Triangle PQR has angle P = 40 degrees and angle Q = 85 degrees. Triangle STU has angle S = 40 degrees and angle U = 55 degrees. Are the triangles similar?
See the answer
Yes
Angle T = 180 - 40 - 55 = 85 degrees, so triangle STU has angles 40, 85, 55 matching triangle PQR's 40, 85, 55 (since R = 180-40-85=55); two pairs of congruent angles means the triangles are similar by AA.
A right triangle has an acute angle of 30 degrees and the side adjacent to it measures 12. What is the length of the hypotenuse?
See the answer
About 13.86 (12 / cos 30°, which is 8√3)
cos(30°) = adjacent/hypotenuse, so hypotenuse = 12 / cos(30°) = 12 / (√3/2) = 24/√3 = 8√3 ≈ 13.86.
Find the center and radius of the circle given by x² + y² - 6x + 4y - 3 = 0.
See the answer
Center (3, -2), radius 4
Completing the square: (x² - 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4, giving (x-3)² + (y+2)² = 16, so center (3,-2) and radius √16 = 4.
A triangle has sides a = 7, b = 9, and included angle C = 50 degrees. What is the length of side c?
See the answer
About 6.9
By the Law of Cosines, c² = 7² + 9² - 2(7)(9)cos(50°) = 49 + 81 - 126(0.643) ≈ 49.98, so c ≈ 7.07; (c ≈ 7.07, rounding may vary slightly).
A game costs 5 dollars to play. You win 20 dollars with probability 0.1, and otherwise win nothing. What is the expected net gain or loss from playing?
See the answer
Expected net loss of 3 dollars
Expected winnings = 0.1(20) + 0.9(0) = 2 dollars; subtracting the 5 dollar cost gives an expected net result of 2 - 5 = -3 dollars.
Questions about this subject
Does this course include formal two-column proofs?
Yes. Proof is central throughout — lines and angles, triangles, parallelograms, circles, and coordinate geometry all involve writing and following logical arguments, though the tutor also accepts paragraph or flow proofs as long as the reasoning is complete.
Is trigonometry in this course the same as a full trig class?
No. This course covers right-triangle trig ratios, the Law of Sines and Cosines, and the complementary angle relationship (sin of an angle equals cos of its complement) — it does not cover trig identities, radians beyond arc length, or the unit circle in depth.
Do I need a graphing calculator or compass and straightedge?
A basic scientific calculator is useful for trig ratios and square roots, and having a compass, straightedge, and protractor on hand helps for the constructions topic, but neither is required for the tutoring sessions themselves.
How does the probability unit connect to the rest of geometry?
It's a separate strand covering sample spaces, conditional probability, counting, and expected value, included because the Common Core traditional pathway places it in the Geometry course; it does not depend on the geometric proof topics.
What if I'm shaky on Algebra I topics like square roots or linear equations?
The tutor will notice if a gap in those skills is blocking a geometry topic — for instance struggling to solve for a variable under a square root in the distance formula — and will work through that piece before continuing.