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Contemplanda

Core mathematics

Geometry

Transformations, similarity, circles, and volume, with two-column proofs the system checks as you write them.

9 units · 61 lessons · 440 practice items

Geometry develops the subject from its foundations through measurement and proof. You work through rigid transformations and congruence, similarity and right-triangle trigonometry, and the relationships that hold inside triangles. Later units cover parallel and perpendicular lines, quadrilaterals and other polygons, circles, and the formulas for circumference, area, and volume, and the course closes with a unit on geometric probability. Each result gives you a tool the next unit assumes.

The recurring format is the two-column proof: you supply the statements and the reasons, and the system checks the logic, which keeps the emphasis on the reasoning behind each step. Other problems have you place vertices on the coordinate plane or solve for a measured length, and most carry hints revealed one step at a time.

Syllabus

Every published lesson in this course, by unit.

Foundations of geometry6 lessons
  1. 1Points, lines, and planes
  2. 2Segments and distance
  3. 3Midpoint and distance in the coordinate plane
  4. 4Angles and angle pairs
  5. 5Basic constructions
  6. 6Reasoning and proof
Transformations and congruence7 lessons
  1. 1Rigid motions and translations
  2. 2Reflections
  3. 3Rotations
  4. 4Symmetry
  5. 5Congruent figures
  6. 6Triangle congruence criteria
  7. 7Congruence proofs and CPCTC
Similarity and right-triangle trigonometry7 lessons
  1. 1Ratios and proportions
  2. 2Similar polygons and dilations
  3. 3Proving triangles similar: AA, SAS, and SSS
  4. 4Parallel lines and proportional parts
  5. 5Right-triangle similarity and the geometric mean
  6. 6Sine, cosine, and tangent
  7. 7Solving right triangles
Parallel and perpendicular lines7 lessons
  1. 1Lines and transversals
  2. 2Parallel lines and transversals
  3. 3Proving lines parallel
  4. 4Slopes of parallel and perpendicular lines
  5. 5Equations of parallel and perpendicular lines
  6. 6Perpendicular lines and distance
  7. 7Unit 4 checkpoint
Relationships within triangles7 lessons
  1. 1Perpendicular and angle bisectors
  2. 2The circumcenter and the incenter
  3. 3Medians and altitudes
  4. 4The triangle midsegment theorem
  5. 5Inequalities in one triangle
  6. 6The hinge theorem
  7. 7Unit 5 checkpoint
Quadrilaterals and other polygons7 lessons
  1. 1Angles of polygons
  2. 2Parallelograms and their properties
  3. 3Proving a quadrilateral is a parallelogram
  4. 4Rectangles, rhombuses, and squares
  5. 5Trapezoids and kites
  6. 6Quadrilaterals in the coordinate plane
  7. 7Unit 6 checkpoint
Circles7 lessons
  1. 1Circles and tangents
  2. 2Arcs, central angles, and chords
  3. 3Inscribed angles and inscribed polygons
  4. 4Angles from chords, secants, and tangents
  5. 5Segment lengths in circles
  6. 6Circles in the coordinate plane
  7. 7Unit 7 checkpoint
Circumference, area, and volume7 lessons
  1. 1Perimeter and area in the plane
  2. 2Circumference and arc length
  3. 3Areas of circles and sectors
  4. 4Areas of regular polygons
  5. 5Surface area and volume of prisms and cylinders
  6. 6Pyramids, cones, and spheres
  7. 7Unit 8 checkpoint
Geometric probability6 lessons
  1. 1What is geometric probability?
  2. 2Probability with lengths
  3. 3Probability with angles and spinners
  4. 4Probability with areas
  5. 5Probability in the coordinate plane
  6. 6Unit 9 checkpoint