# High School Geometry

**Relevant Brilliant Courses (Online Content)**

The four courses below are the foundations for all of the mathematics offered on Brilliant. If you’re an educator with a group of 10 or more students you want to give full access to these courses, contact pricing@brilliant.org to learn more about Brilliant’s discounts for school groups.

**Deep Diving Math Enrichment Problem Sets (Printable Content)**

Too often, school math is all about “racing to finish” instead of diving deep to understand and explore creative, tangential lines of inquiry. These sets were written to inspire deep diving exploration that extends and enriches the core mathematical topics and skills introduced in the geometry common core curriculum.

Each of the “Practice-Challenge-Culmination” problem sets listed below takes some foundational skill in the common core curriculum for that grade and, after a few practice problems, extends the concept to more creative challenges, and then to a single, deep-dive question.

#### Contents

## Quadrilaterals and Other Polygons

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentEXPLORING QUADRILATERALS [Printable PDF] CCSS.MATH.CONTENT.HSG.CO.C.11 CCSS.MATH.CONTENT.HSG.SRT.B.5 This activity aims to improve students’ problem solving skills by exploring several different types of problems involving quadrilaterals. Students begin by identifying types of quadrilaterals based on clues. Then, students apply facts about quadrilateral angle relationships and similar triangles. In the culmination, creative problem solving allows for an elegant solution as students aim to find the area of a quadrilateral created by two overlapping squares. Out Of The Box Geometry: Is It Regular? PROOFS: PEGBOARD RECTANGLES [Printable PDF] CCSS.MATH.CONTENT.HSA.CED.A.2 CCSS.MATH.CONTENT.HSG.MG.A.3 This tightly sequenced problem set provides students with an opportunity to explore Pick’s theorem, a theorem involving the area of lattice polygons that is widely admired for its elegance and simplicity. The set has a geometry proof of an algebraic nature: the case of Pick's theorem that involves only unit squares. It is a structured exploration in which problems rely on answers to previous problems. Because of the structure, students should be exposed to the culmination question before they start the problems. Out Of The Box Geometry: Pegboard Triangles REGULAR POLYGONS [Printable PDF] CCSS.MATH.CONTENT.HSG.SRT.B.4 CCSS.MATH.CONTENT.HSG.SRT.B.5 CCSS.MATH.CONTENT.HSG.SRT.C.8 In this activity, students extend their understanding of regular polygons by decomposing figures into congruent pieces and right triangles. Students apply properties of symmetry, congruence, and special right triangles to calculate and compare areas and perimeters. Out Of The Box Geometry: Triangles and Hexagons

## Lines and Angle Relationships

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentANGLES WITH PARALLEL LINES [Printable PDF] CCSS.MATH.CONTENT.HSG.CO.C.9 CCSS.MATH.CONTENT.HSG.CO.C.11 CCSS.MATH.CONTENT.HSA.REI.B.3 This set of angle-solving problems begins with standard, parallel-line angle hunting before moving into more creative applications. After examining not just angles, but also shapes, involving parallel lines, students set out to prove whether or not the midpoints of the four sides of any simple quadrilateral make a parallelogram. Out of the Box Geometry: Angle Hunting Axioms ANGLE HUNTING IN LINES AND TRIANGLES [Printable PDF] CCSS.MATH.CONTENT.HSG.CO.C.9 CCSS.MATH.CONTENT.HSG.SRT.A.2 CCSS.MATH.CONTENT.HSG.CO.A.5 This angle hunting activity requires creative approaches to using vertical angles, linear pairs, the triangle sum theorem, and corresponding angles. Students begin by angle hunting through mazes of triangles and parallel lines. Then, students move on to more challenging angle hunts that are solved most elegantly through creative applications of rotations and translations. In the culmination, students aim to find the sum of three angles (without using trigonometry) given only lines drawn on a grid. Out of the Box Geometry: Angle Hunting Axioms

## Area, Surface Area, and Volume

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentCOMPOSITE GEOMETRY [Printable PDF] CCSS.MATH.CONTENT.HSG.CO.B.6 CCSS.MATH.PRACTICE.MP1 CCSS.MATH.PRACTICE.MP7 These problems explore the geometry of composite figures while providing students the opportunity to be creative with their problem solving strategies. Students begin by exploring composite figures that can be broken into triangles and rectangles before working with composite figures that contain circles. The culmination provides one final composite figure challenge that can be solved with some long calculations or some very elegant shortcuts. Mathematical Fundamentals: Challenging Composites

## Triangle Relationships

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentPROOF: ANGLE BISECTOR THEOREM [Printable PDF] CCSS.MATH.CONTENT.HSG.SRT.B.4 CCSS.MATH.CONTENT.HSG.SRT.B.5 This series of problems leads to a proof of the angle bisector theorem, and then a tricky problem that applies it! The intent for the problems is for the students to produce proofs rather than just answers. The full angle bisector theorem should be extracted from the second challenge problem before moving to the (very tricky!) culmination problem. Outside of the Box Geometry: Cevians TRIANGLE CLASSIFICATION [Printable PDF] CCSS.MATH.CONTENT.HSG.SRT.B.4 CCSS.MATH.CONTENT.HSG.SRT.C.8 This series includes some creative problems involving identifying the different triangle types and using them in context. The harder problems involve finding different triangle types on a pegboard and on a cube. Outside of the Box Geometry: Pegboard Triangles TRIANGLE CONGRUENCE [Printable PDF] CCSS.MATH.CONTENT.HSG.SRT.B.4 CCSS.MATH.CONTENT.HSG.SRT.B.5 CCSS.MATH.CONTENT.HSG.SRT.C.8 This series provides multiple contexts in which students can apply their understanding of ASA, SSS, and AAS. Students also examine the two possible triangles in the SSA case. Finally, students hunt for similar and congruent triangles in a puzzle of squares and right triangles as they attempt to prove whether a statement is true or false. Outside of the Box Geometry: Congruence and Similarity TRIANGLE SIMILARITY [Printable PDF] CCSS.MATH.CONTENT.HSG.SRT.A.2 CCSS.MATH.CONTENT.HSG.SRT.A.3 CCSS.MATH.CONTENT.HSG.SRT.B.5 These problems deepen a student’s understanding of similar triangles by covering multiple perspectives on triangle similarity. The problems include identifying similar triangles and calculating side lengths and areas of similar triangles. In the culmination, students have a chance to explore how similarity connects to fractals. Mathematical Fundamentals: Similarity CEVIANS [Printable PDF] CCSS.MATH.CONTENT.HSG.CO.C.10 CCSS.MATH.CONTENT.HSG.SRT.C.8 CCSS.MATH.CONTENT.HSG.SRT.B.5 In this activity, students get a deep look at cevians and how they’re related. After answering conceptual questions about altitudes, medians, angle bisectors, and perpendicular bisectors, students dive into more challenging applications of cevians. In the culmination, students determine a ratio between segments of medians and side lengths in an equilateral triangle and have an opportunity to begin down the path of proving Stewart’s theorem. Outside of the Box Geometry: Cevians

## Similarity, Congruence, and Transformations

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentSIMILARITY SCALING [Printable PDF] CCSS.MATH.CONTENT.HSG.SRT.A.2 CCSS.MATH.CONTENT.HSG.SRT.B.5 These similarity problems explore how scaling figures affects their areas and volumes. Students deepen their understanding of the impacts of scaling by exploring abstract figures. In the culmination, students look for patterns among similar figures to determine a square’s area. Mathematical Fundamentals: Scaling

## Right Triangles and Trigonometry

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentPYTHAGOREAN THEOREM PROOFS [Printable PDF] CCSS.MATH.CONTENT.HSG.SRT.B.4 CCSS.MATH.CONTENT.HSG.SRT.B.5 This special series of problems explores four different proofs of the Pythagorean theorem. It includes extensive use of algebra and similarity. The initial problems are designed to guide students through the logic involved in the proofs. The culmination encourages students to develop their own proof based on a given diagram. Outside of the Box Geometry: Right Triangles

## Coordinate Geometry

Printable PDFsCommon Core StandardsDescriptionRelated Course ContentCOORDINATE GEOMETRY [Printable PDF] CCSS.MATH.CONTENT.HSG.GPE.B.5 CCSS.MATH.CONTENT.HSG.GPE.B.7 In this activity, students deepen their understanding of the algebra-geometry connection as they explore midpoints, distances, slopes, and areas of polygons. After working through midpoint and distance scenarios, students identify coordinates and write equations of lines to create polygons. In the culmination, students combine all of their coordinate geometry skills to find the area of a quadrilateral created from a line, a point, and a circle. Mathematical Fundamentals: Distance on a Graph EQUATIONS OF CIRCLES [Printable PDF] CCSS.MATH.CONTENT.HSG.GPE.A.1 This activity strives to deepen the algebra-geometry connection as students explore the algebraic representation of circles. After examining sets of equations and graphs, students explore tangent lines and shortest distances between circles. In the culmination, students derive their own formula for the equation of a circle using the Pythagorean theorem. Mathematical Fundamentals: Circles

## Properties of Circles

Coming Soon

**Cite as:**High School Geometry.

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