Math Ladder · High school · Grade 10

Geometry

Reasoning and proof, congruence and similarity, right-triangle trigonometry, circles, coordinates, area and volume.

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64 topics in 10 units. Builds on Algebra 1. Leads to Algebra 2. It also opens the way to Physics 1 on the Physics Ladder, and Weather and Climate and Geology on the Earth & Space Ladder. Not sure this is the right course? Find your rung.

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Unit 1 Tools of Geometry

6 topics

Undefined terms and definitions, distance and midpoints, and compass-and-straightedge constructions.

  1. Points, Lines and Planes
  2. Distance and Midpoint Formulas
  3. Angles and Angle Bisectors
  4. Copying and Bisecting with Compass and Straightedge
  5. Constructing Perpendicular and Parallel Lines
  6. Constructing Polygons in a Circle

Unit 2 Reasoning and Proof

4 topics

Use if-then logic and the properties of equality to write proofs about lines and angles.

  1. If-Then Statements
  2. Properties of Equality as Reasons
  3. Writing a Proof
  4. Proving Theorems about Lines and Angles

Unit 3 Rigid Motions

4 topics

Define transformations precisely, compose them, and find symmetries.

  1. Transformations as Functions
  2. Precise Definitions of Rigid Motions
  3. Symmetries of Polygons
  4. Composing Transformations

Unit 4 Triangle Congruence

5 topics

Prove triangles congruent and use congruent parts.

  1. Congruence and Rigid Motions
  2. Corresponding Parts
  3. SSS, SAS and ASA
  4. AAS and Hypotenuse-Leg
  5. Proofs with Congruent Triangles

Unit 5 Properties of Triangles

5 topics

Prove triangle theorems and find the special points of a triangle.

  1. Triangle Sum and Exterior Angle Proofs
  2. Isosceles and Equilateral Triangles
  3. Perpendicular Bisectors and the Circumcenter
  4. Incenter, Centroid and Orthocenter
  5. Triangle Midsegments

Unit 6 Quadrilaterals and Polygons

5 topics

Find polygon angle sums and prove properties of parallelograms and special quadrilaterals.

  1. Polygon Angle Sums
  2. Properties of Parallelograms
  3. Proving a Quadrilateral Is a Parallelogram
  4. Rectangles, Rhombuses and Squares
  5. Proofs with Coordinates

Unit 7 Similarity

8 topics

Use dilations to define similarity, prove triangles similar, and use proportional segments.

  1. Properties of Dilations
  2. Similarity Through Transformations
  3. Proving AA Similarity
  4. SSS and SAS Similarity
  5. The Triangle Proportionality Theorem
  6. Dividing a Segment in a Ratio
  7. Similar Right Triangles and the Geometric Mean
  8. Proving the Pythagorean Theorem with Similarity

Unit 8 Right Triangles and Trigonometry

9 topics

Use special right triangles and the trig ratios to solve triangles.

  1. 45-45-90 and 30-60-90 Triangles
  2. Sine, Cosine and Tangent
  3. Sine and Cosine of Complementary Angles
  4. Solving Right Triangles
  5. Finding Angles with Inverse Trig
  6. Angles of Elevation and Depression
  7. Triangle Area with Sine
  8. The Law of Sines
  9. The Law of Cosines

Unit 9 Circles

10 topics

Learn arcs, angles, chords and tangents, arc length and sector area, and circle equations.

  1. Parts of a Circle
  2. Central Angles and Arcs
  3. Inscribed Angles
  4. Tangent Lines
  5. Inscribed Quadrilaterals
  6. Arc Length
  7. Sector Area
  8. Equation of a Circle
  9. Finding the Center by Completing the Square
  10. Parabola from Focus and Directrix

Unit 10 Area, Volume and Modeling

8 topics

Explain the volume formulas, find surface areas, scale solids, and model real objects.

  1. Area of Regular Polygons
  2. Perimeter and Area on the Coordinate Plane
  3. Cavalieri's Principle
  4. Volume of Pyramids, Cones and Spheres
  5. Surface Area of Solids
  6. Cross Sections and Solids of Revolution
  7. Scaling Area and Volume
  8. Modeling, Density and Design
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