Geometry | Index

Geometry is the study of shapes, angles, distance and space. You’ll also learn to prove facts about them.

The course starts with lines, angles, proofs and transformations. It goes on to triangles, polygons, similarity, right triangle trig, circles, and surface area and volume.

It’s for high school Geometry students and anyone preparing for a geometry test.

What to know first: Algebra 1, especially solving equations, slope and simplifying square roots. Basic area and circle formulas are in Pre-Algebra.

Every lesson in this course is live. The list below shows them all in teaching order.

Units in this course

Unit 1: Points, Lines, Planes and Angles

Practice pack for this unit: Geometry Unit 1 Practice Pack. It has 48 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Points, Lines and Planes — the building blocks of geometry and how to name them
  2. Segment Addition Postulate — measure segments and use congruent segments
  3. Midpoint Formula — find the midpoint of a segment on the coordinate plane
  4. Distance Formula — find the length of a segment from its endpoints
  5. Partitioning a Line Segment — find the point that divides a segment in a given ratio
  6. Measuring and Classifying Angles — measure angles, name their types and use the angle addition postulate
  7. Angle Pairs — complementary, supplementary and vertical angles and linear pairs
  8. Basic Constructions — copy and bisect segments and angles with a compass and straightedge

Unit 2: Reasoning and Proof

Practice pack for this unit: Geometry Unit 2 Practice Pack. It has 51 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Conditional Statements — write converses, inverses and contrapositives
  2. Inductive and Deductive Reasoning — make conjectures from patterns and reach conclusions with logic
  3. Algebraic Proofs — justify each step of solving an equation with properties of equality
  4. Two-Column Proofs — write proofs about segments and angles with statements and reasons
  5. Proving Angle Relationships — prove the vertical angles theorem and other angle pair theorems

Unit 3: Parallel and Perpendicular Lines

Practice pack for this unit: Geometry Unit 3 Practice Pack. It has 42 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Parallel Lines and Transversals — corresponding, alternate interior and same-side interior angles
  2. Proving Lines Parallel — use angle pairs to show that two lines are parallel
  3. Proofs with Perpendicular Lines — theorems about perpendicular lines and the distance from a point to a line

Also see: slopes and equations of parallel and perpendicular lines in Algebra 1, Unit 4.

Unit 4: Transformations

Practice pack for this unit: Geometry Unit 4 Practice Pack. It has 41 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Translations — slide a figure and describe the move with coordinates
  2. Reflections — flip a figure over a line
  3. Rotations — turn a figure around a point
  4. Compositions of Transformations — combine rigid motions and describe the result
  5. Dilations — enlarge or shrink a figure by a scale factor
  6. Symmetry — line symmetry and rotational symmetry

Unit 5: Triangle Congruence

Practice pack for this unit: Geometry Unit 5 Practice Pack. It has 50 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Triangle Angle Sum Theorem — the angles of a triangle add to 180 degrees, plus the exterior angle theorem
  2. Congruent Figures — use rigid motions and matching parts to show figures are congruent
  3. SSS and SAS Congruence — prove triangles congruent with three sides, or two sides and the angle between
  4. ASA, AAS and HL Congruence — prove triangles congruent with angle-side combinations and right triangles
  5. CPCTC — show that matching parts are congruent once the triangles are
  6. Isosceles and Equilateral Triangles — the base angles theorem and its converse

Unit 6: Relationships Within Triangles

Practice pack for this unit: Geometry Unit 6 Practice Pack. It has 43 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Perpendicular and Angle Bisectors — properties of points on perpendicular bisectors and angle bisectors
  2. Circumcenter and Incenter — where a triangle’s bisectors meet and what each point does
  3. Medians and Altitudes of Triangles — the centroid and orthocenter, and the centroid’s 2-to-1 ratio
  4. Triangle Midsegment Theorem — the segment joining two midpoints is parallel to the third side and half as long
  5. Triangle Inequality Theorem — decide whether three lengths can make a triangle, and order sides and angles
  6. Hinge Theorem — compare sides and angles in two triangles

Unit 7: Polygons and Quadrilaterals

Practice pack for this unit: Geometry Unit 7 Practice Pack. It has 42 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Interior and Exterior Angles of Polygons — find angle sums and single angles of any polygon
  2. Properties of Parallelograms — sides, angles and diagonals of parallelograms
  3. Proving a Quadrilateral Is a Parallelogram — the conditions that prove a shape is a parallelogram
  4. Rectangles, Rhombuses and Squares — properties of special parallelograms and their diagonals
  5. Trapezoids and Kites — properties of trapezoids, isosceles trapezoids and kites
  6. Coordinate Proofs — prove properties of figures placed on the coordinate plane

Unit 8: Similarity

Practice pack for this unit: Geometry Unit 8 Practice Pack. It has 41 pages of practice problems, with a step-by-step answer key and a cheat sheet.

  1. Similar Polygons — scale factors and similarity transformations
  2. AA Similarity — prove triangles similar with two pairs of congruent angles
  3. SSS and SAS Similarity — prove triangles similar with proportional sides
  4. Triangle Proportionality Theorem — a line parallel to one side splits the other two sides proportionally
  5. Triangle Angle Bisector Theorem — an angle bisector divides the opposite side in the ratio of the other two sides

Unit 9: Right Triangles and Trigonometry

  1. Pythagorean Theorem — find a missing side of a right triangle
  2. Converse of the Pythagorean Theorem — use the sides to tell whether a triangle is right, acute or obtuse
  3. Special Right Triangles — side ratios in 45-45-90 and 30-60-90 triangles
  4. Similar Right Triangles — the altitude to the hypotenuse and the geometric mean
  5. Trigonometric Ratios — sine, cosine and tangent in right triangles (SOH CAH TOA)
  6. Solving Right Triangles — find missing sides and angles with trig ratios and inverse trig
  7. Angles of Elevation and Depression — solve height and distance problems with right triangle trig

Also see: the law of sines and law of cosines in Trigonometry, Unit 8.

Unit 10: Circles

  1. Parts of a Circle — radius, diameter, chords, secants and tangents
  2. Tangents to a Circle — why a tangent meets the radius at a right angle, and tangent segment rules
  3. Arcs and Central Angles — measure arcs and use central angles
  4. Inscribed Angles — inscribed angles, intercepted arcs and shapes inscribed in circles
  5. Angle Relationships in Circles — angles formed by chords, secants and tangents
  6. Segment Relationships in Circles — lengths of intersecting chords, secants and tangents
  7. Arc Length — find the length of an arc from its central angle
  8. Area of a Sector — find the area of a slice of a circle
  9. Equation of a Circle — write and graph a circle from its center and radius

Unit 11: Area, Surface Area and Volume

  1. Area of Regular Polygons — use the apothem and perimeter to find the area of a regular polygon
  2. Cross Sections — shapes made by slicing solids, and solids formed by rotating shapes
  3. Surface Area of Prisms and Cylinders — add the areas of all faces and curved surfaces
  4. Surface Area of Pyramids and Cones — use slant height to find surface area
  5. Volume of Prisms and Cylinders — multiply the area of the base by the height
  6. Volume of Pyramids and Cones — find volume as one-third of base area times height
  7. Surface Area and Volume of Spheres — use the radius to find a sphere’s surface area and volume
  8. Similar Solids — how a scale factor changes surface area and volume

Also see: Area of Triangles and Circumference and Area of Circles (Pre-Algebra, Unit 6).

For Teachers: Standards Alignment

Lessons follow the Common Core high school Geometry standards: congruence, similarity, right triangle trig, circles, coordinate geometry, measurement and modeling. Slopes of parallel and perpendicular lines are taught in Algebra 1. The laws of sines and cosines are in Trigonometry, and probability is in Statistics.

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