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Course

10th Grade Math: Geometry

Write proofs, use transformations, prove triangles congruent and similar, solve right triangles with trigonometry, and study circles and solids.

Work through the lessons in order, from points and lines to independent problem-solving. By the end, you can use logic and constructions, prove theorems about lines and angles, and describe congruence with rigid motions. You can prove triangles congruent, use properties of triangles and quadrilaterals, and prove triangles similar with dilations. You can solve right triangles with the Pythagorean Theorem and trigonometry and use the Laws of Sines and Cosines. You can find arc lengths, sector areas, and inscribed angles and write the equation of a circle. You can write coordinate proofs and find volumes of prisms, cylinders, pyramids, cones, and spheres.

How to use these lessons

  • Start at the lesson marked Do this. Go on when you can do the line under the title by yourself.
  • Play the Games first. Then take the Quizzes. Eight of ten correct passes. The Worksheets are on paper. The answers are at the bottom.
  • Each lesson builds on the one before it. If a lesson is hard, go back one.
  • Draw and label a diagram for every problem. Keep a compass, a straightedge, and a protractor nearby. In a proof, give a reason for every statement.

Each lesson has games, quizzes, and worksheets.

0 of 43

Foundations, Constructions, and Logic

  1. 1Points, Lines, and PlanesName and describe points, lines, rays, segments, and planes.
    • Use correct notation for lines, rays, and segments.
    • Know the undefined terms of geometry.
    • Describe how lines and planes intersect.
  2. 2Distance, Midpoint, and Angle MeasureFind lengths, midpoints, and angle measures.
    • Use the distance formula on the coordinate plane.
    • Use the midpoint formula.
    • Use the segment and angle addition postulates.
  3. 3ConstructionsConstruct copies, bisectors, and perpendiculars with a compass and straightedge.
    • Copy a segment and an angle.
    • Bisect a segment and an angle.
    • Construct a perpendicular and a parallel line.
  4. 4Logic and Conditional StatementsWrite conditionals, converses, and counterexamples.
    • Write the converse, inverse, and contrapositive.
    • Find a counterexample to a false statement.
    • Tell inductive from deductive reasoning.

Lines, Angles, and Proofs

  1. 5Parallel Lines and TransversalsUse angle pairs formed by parallel lines and a transversal.
    • Name corresponding, alternate interior, and same-side angles.
    • Find unknown angles.
    • Prove lines are parallel from angle pairs.
  2. 6Write ProofsWrite two-column and paragraph proofs.
    • Use definitions, postulates, and theorems as reasons.
    • Use algebra properties in proofs.
    • Write a proof in two-column and paragraph form.
  3. 7Prove Line and Angle TheoremsProve theorems about vertical angles and perpendicular bisectors.
    • Prove vertical angles are congruent.
    • Prove points on a perpendicular bisector are equidistant from the endpoints.
    • Prove the alternate interior angles theorem.

Transformations and Congruence

  1. 8Rigid MotionsDescribe translations, reflections, and rotations as functions.
    • Write rules such as (x, y) → (−y, x).
    • Describe a sequence of transformations.
    • Know rigid motions keep lengths and angles.
  2. 9Congruence Through Rigid MotionsShow figures are congruent with a sequence of rigid motions.
    • Define congruence with rigid motions.
    • Find a sequence that maps one figure to another.
    • Name corresponding parts.
  3. 10SymmetryFind line and rotational symmetry in figures.
    • Find the lines of symmetry of a polygon.
    • Find the angle of rotational symmetry.
    • Describe transformations that map a figure onto itself.

Triangles

  1. 11Triangle CongruenceProve triangles congruent with SSS, SAS, ASA, AAS, and HL.
    • Choose the right congruence criterion.
    • Know why SSA does not work.
    • Use CPCTC in proofs.
  2. 12Triangle Angle TheoremsProve and use the triangle sum and exterior angle theorems.
    • Prove the angles of a triangle add to 180°.
    • Use the exterior angle theorem.
    • Find unknown angles with algebra.
  3. 13Isosceles and Equilateral TrianglesUse the base angles theorem and its converse.
    • Know base angles of an isosceles triangle are equal.
    • Use the converse to find equal sides.
    • Know an equilateral triangle is equiangular.
  4. 14Bisectors, Medians, and AltitudesFind the circumcenter, incenter, centroid, and orthocenter.
    • Construct each triangle center.
    • Know the centroid divides a median 2 to 1.
    • Use the midsegment theorem.
  5. 15Triangle InequalitiesDecide if three lengths make a triangle and order its sides and angles.
    • Use the triangle inequality theorem.
    • Know the longest side is opposite the largest angle.
    • Find the range of possible lengths for the third side.

Polygons and Quadrilaterals

  1. 16Polygon Angle SumsFind interior and exterior angle sums of polygons.
    • Use (n − 2) × 180° for the interior sum.
    • Know exterior angles add to 360°.
    • Find each angle of a regular polygon.
  2. 17Properties of ParallelogramsProve and use properties of parallelograms.
    • Prove opposite sides and angles are congruent.
    • Prove the diagonals bisect each other.
    • Show a quadrilateral is a parallelogram.
  3. 18Rectangles, Rhombuses, and SquaresUse the properties of special parallelograms.
    • Know rectangle diagonals are congruent.
    • Know rhombus diagonals are perpendicular.
    • Classify a quadrilateral from its properties.
  4. 19Trapezoids and KitesUse the properties of trapezoids and kites.
    • Use the trapezoid midsegment theorem.
    • Know the properties of an isosceles trapezoid.
    • Know kite diagonals are perpendicular.

Similarity

  1. 20DilationsDilate figures and describe the effect of the scale factor.
    • Dilate from a center with scale factor k.
    • Know a dilation maps a line to a parallel line.
    • Define similarity with dilations and rigid motions.
  2. 21Similar TrianglesProve triangles similar with AA, SAS, and SSS.
    • Use the AA similarity criterion.
    • Write and solve proportions for missing sides.
    • Solve indirect measurement problems with shadows.
  3. 22Proportions in TrianglesUse the side-splitter and angle bisector theorems.
    • Use the side-splitter theorem.
    • Use the triangle angle bisector theorem.
    • Prove the Pythagorean Theorem with similar triangles.

Right Triangles and Trigonometry

  1. 23The Pythagorean Theorem and Its ConverseUse the Pythagorean Theorem and classify triangles by their sides.
    • Find missing sides in right triangles.
    • Use the converse to test for a right triangle.
    • Classify triangles as acute, right, or obtuse.
  2. 24Special Right TrianglesFind sides of 45-45-90 and 30-60-90 triangles.
    • Use the ratio 1 : 1 : √2.
    • Use the ratio 1 : √3 : 2.
    • Leave exact answers with radicals.
  3. 25Sine, Cosine, and TangentFind trigonometric ratios in a right triangle.
    • Use SOH CAH TOA.
    • Know the ratios come from similar triangles.
    • Know sin A = cos (90° − A).
  4. 26Solve Right TrianglesFind missing sides and angles with trigonometry.
    • Use inverse trig to find angles.
    • Solve angle of elevation and depression problems.
    • Round answers sensibly.
  5. 27The Laws of Sines and CosinesSolve any triangle with the Law of Sines or the Law of Cosines.
    • Use the Law of Sines.
    • Use the Law of Cosines.
    • Find area with 1/2 ab sin C.

Circles

  1. 28Circle Parts and ArcsName parts of a circle and find arc measures.
    • Name radii, chords, diameters, secants, and tangents.
    • Find minor and major arc measures.
    • Know all circles are similar.
  2. 29Arc Length and RadiansFind arc length and convert between degrees and radians.
    • Find arc length as a fraction of the circumference.
    • Define a radian.
    • Convert between degrees and radians.
  3. 30Sector AreaFind the area of a sector.
    • Find sector area as a fraction of the circle's area.
    • Find the area of a segment of a circle.
    • Solve pizza and sprinkler problems.
  4. 31Inscribed AnglesUse the relationship between inscribed angles and arcs.
    • Know an inscribed angle is half its arc.
    • Know an angle inscribed in a semicircle is a right angle.
    • Know opposite angles of an inscribed quadrilateral add to 180°.
  5. 32Tangents, Chords, and SecantsUse properties of tangents, chords, and secants.
    • Know a tangent is perpendicular to the radius.
    • Use segment products for chords and secants.
    • Find angles formed by chords, secants, and tangents.
  6. 33Equations of CirclesWrite and graph the equation of a circle.
    • Use (x − h)² + (y − k)² = r².
    • Find the center and radius by completing the square.
    • Decide if a point lies on a circle.

Coordinate Geometry

  1. 34Coordinate ProofsProve geometric facts with coordinates.
    • Place figures on the coordinate plane wisely.
    • Use slope and distance to classify a quadrilateral.
    • Prove a midsegment is parallel to a side.
  2. 35Partition a SegmentFind the point that divides a segment in a given ratio.
    • Find the point 2/5 of the way from A to B.
    • Divide a segment in the ratio 1 : 3.
    • Connect partitioning to dilations.
  3. 36Perimeter and Area with CoordinatesFind the perimeter and area of polygons on the coordinate plane.
    • Use the distance formula for perimeter.
    • Find area by decomposing or enclosing in a rectangle.
    • Find the area of a triangle from its vertices.

Volume and Modeling

  1. 37Volume of Prisms and CylindersFind and explain volumes of prisms and cylinders.
    • Use V = Bh.
    • Explain Cavalieri's principle.
    • Find the volume of oblique solids.
  2. 38Pyramids, Cones, and SpheresFind the volume and surface area of pyramids, cones, and spheres.
    • Use V = 1/3 Bh.
    • Use V = 4/3 πr³ and SA = 4πr².
    • Solve composite solid problems.
  3. 39Cross Sections and Solids of RevolutionDescribe cross sections and the solid made by rotating a shape.
    • Name cross sections of prisms, cylinders, and cones.
    • Describe the solid made by rotating a rectangle or triangle.
    • Connect 2D shapes to 3D solids.
  4. 40Density and Design ModelingUse geometry to model density and design problems.
    • Find population density per square mile.
    • Find the mass of an object from its density.
    • Design a box to meet a constraint.

Review, Mixed Practice, and Challenges

  1. 41Proof and Congruence ReviewSolve mixed problems with proofs, transformations, triangles, and quadrilaterals.
    • Write proofs with reasons.
    • Prove triangles congruent and similar.
    • Use properties of quadrilaterals.
  2. 42Trigonometry, Circle, and Volume ReviewSolve mixed problems with trigonometry, circles, and solids.
    • Solve right and oblique triangles.
    • Find arcs, sectors, and circle equations.
    • Find volumes of solids.
  3. 43Geometry ChallengeComplete mixed questions covering the major units.
    • Solve problems from every unit without hints.
    • Explain each answer with a theorem.
    • Find the areas that need more practice.