Geometry

The concept of a rigorous mathematical proof is introduced. Students begin by exploring proof fundamentals, such as postulates, definitions, theorems, and the properties of congruence. Once mastered, they will use these ideas to prove important geometric theorems. Opportunities to demonstrate and develop their proof-writing skills are ongoing throughout this course.

Students will explore geometric transformations in the plane. They will construct functions that represent specific mappings and apply these functions to manipulate various geometric objects.

The course introduces the concept of rigid motions as transformations that preserve both distance and angle. Students will learn to identify whether a given transformation qualifies as a rigid motion and describe the concept of congruence in terms of rigid motions. Additionally, students will master the congruence criteria for triangles and use them to prove mathematical statements.

Progressing from the concept of congruence, students explore the notion of similarity. They will learn to describe similarity through similarity transformations and apply similarity concepts to solve problems involving similar polygons. Students will proficiently use the AA, SSS, and SAS similarity criteria and construct formal proofs about similar triangles.

Students will undertake a deep exploration of circles. They begin by studying circle fundamentals such as circumference, area, and circular sectors. Various circle theorems are introduced, including the inscribed angle theorem, Thales' theorem, and the tangent-chord theorem. Students will solve problems involving secant and tangent lines to circles, explore the concept of radian measure, and construct formal proofs demonstrating that any two circles are similar.

Much of this course is dedicated to extending students' knowledge to problems within the coordinate plane. They will derive equations of parallel and perpendicular lines, calculate distances and midpoints in the plane, partition line segments, and apply their knowledge to compute areas and perimeters of polygons. Students will transfer their knowledge of circle theorems to coordinate plane problems.

A key objective of this course is to deepen students' understanding of triangles. Students will become proficient in applying the Pythagorean theorem, understand its role in deducing properties of special right triangles, and use this knowledge to solve more complex problems. Students will have the opportunity to prove multiple triangle theorems. Moreover, they will encounter trigonometric ratios, solve problems requiring computing angles and side lengths of right triangles, and apply these skills to more complex scenarios.

Students will expand their geometric knowledge beyond the plane, exploring both polyhedra and non-polyhedra. They will learn to calculate volumes and surface areas of multiple shapes and solve related problems. Concepts such as Euler's formula for polyhedra, the Platonic solids, and volumes of revolution will also be introduced.

Finally, students develop their understanding of probability and combinatorics. They will explore events, Venn diagrams, unions and intersections of events, the addition law, conditional probability, tree diagrams, permutations, and combinations. Throughout this unit, students will practice applying these concepts to model real-world problems.

Congruence, Similarity, & Proof

• Proving the vertical angles theorem.
• Proving the alternate and consecutive interior and exterior angle theorems and their converses.
• Proving the linear pair perpendicular theorem, the perpendicular bisector theorem, and their converses.
• Applying translations, reflections, dilations, stretches, and composite transformations to two-dimensional figures.
• Using functions to represent transformations in the coordinate plane and using them to find the image of an object.
• Understanding and applying the ASA, AAS, SAS, SSS, and HL congruence criteria.
• Explaining how rigid motions can be used to demonstrate that two figures are congruent.
• Understanding and applying the reflexive, symmetric, and transitive properties of congruence.
• Proving that two triangles are congruent and applying CPCTC.
• Polygon similarity, including working with area scale factors and explaining how similarity transformations are used to show that two polygons are similar.
• Understanding and applying the AA, SSS, and SAS similarity criteria for triangles.
• Proving that two triangles are similar, and applying CPSTP and corresponding angles in similar triangles.
• Understanding and applying the midpoint and triangle proportionality theorems.
• Proving the midpoint theorem.

Circles & Radian Measure

• Circle fundamentals: Radii, diameters, chords, interiors and exteriors, circumference, arcs, sectors, and segments.
• Calculating circumferences and areas of circles, circular sectors, and solving related problems.
• Solving problems involving inscribed angles, including applying the inscribed angle theorem, Thales' theorem, and problems involving inscribed quadrilaterals.
• Solving problems with tangent lines to circles, tangent-tangent, secant-sectant, and sectant-tangent lines, and applying the tangent-chord theorem.
• Proving that all circles are similar.
• Understanding radians, converting between radians and degrees, and using radians to calculate arc lengths and areas of sectors.

Geometry in the Coordinate Plane

• Working with parallel and perpendicular lines in the coordinate plane.
• The distance formula, its derivation, and how to use it to find the shortest distance between a point and a line.
• Partitioning a directed line segment, solving related problems, and applying the midpoint formula.
• Calculating perimeters and areas of polygons in the coordinate plane.
• Finding equations of circles in the coordinate plane.
• Determining a circle's center and radius by completing the square.
• Calculating intercepts of circles and finding intersections of circles and lines.
• Finding equations of tangent lines to circles.
• Using the properties of the perpendicular bisectors of diameters and chords to solve problems in the coordinate plane.
• Applying Thales' theorem in the coordinate plane.

Triangles, Quadrilaterals, & Trigonometry

• Understanding, applying, and proving the exterior angle theorem.
• Understanding and applying the Pythagorean Theorem and its converse.
• The 45-45-90 and 30-60-90 special right triangles
• Calculating areas of 45-45-90, 30-60-90, and equilateral triangles.
• Proving the sum of the interior angles of a triangle equals 180 degrees.
• Proving the isosceles triangle theorem, its converse, and other results related to isosceles triangles.
• Understanding medians and centroids of triangles, applying the centroid ratio theorem, and finding the coordinates of a triangle's centroid.
• Prove the centroid and centroid ratio theorems and derive the formula for the coordinates of a triangle's centroid.
• Proving properties of parallelograms and theorems relating parallelograms and rectangles.
• Using trigonometric ratios to calculate side lengths and angle measures in right triangles.
• The reciprocal trigonometric ratios.
• Recalling special trigonometric ratios and their derivations.
• Understanding the relationship between trigonometric ratios in similar right triangles.
• Calculating areas in right triangles using trigonometry.
• Complementary angles in right triangles and the relation between trigonometric functions and their cofunctions.
• Using trigonometry to model and solve real-world and mathematical problems, including cases with multiple right triangles.

Solid Geometry

• Identifying three-dimensional figures and correctly applying the terminology associated with three-dimensional geometry.
• Calculating surface areas using nets.
• Applying the distance formula in three dimensions.
• Applying Euler's formula for polyhedra and understanding the significance of the five Platonic Solids.
• Working with surface areas and volumes of similar solids.
• Calculating surface areas and volume for various three-dimensional figures, including cubes, rectangular solids, pyramids, cylinders, cones, spheres, and canonical frustums.
• Calculating volumes of revolution.

Probability & Combinatorics

• Essential concepts from probability, including sample spaces, event complements, compound events, and Venn diagrams,
• Calculating probabilities involving unions and intersections of events and using these ideas to solve real-world problems.
• Applying the addition law and understanding mutually exclusive events.
• Conditional probabilities, including the multiplication law and its derivation, the law of total probability, independent events, and tree diagrams.
• Applying the rule of sum and the rule of product.
• Determining the number of ways to order objects and calculating permutations and combinations.
• Calculating probabilities using combinatorics.