Students investigate how geometric claims become trustworthy by using constructions, theorem-based reasoning, and formal proof to determine unknown lengths and angle measures in complex figures. Across the project, they build and revise courtroom-style arguments with vertical angles, transversals, perpendicular bisectors, triangle relationships, special right triangles, and circle-based diagrams, communicating their reasoning orally and in writing. The experience culminates in public case exhibits and proof portfolios that show students can justify conclusions precisely, respond to critique, and strengthen an argument with evidence.

Students will use vertical angles, transversals, perpendicular bisectors, triangle relationships, and special right triangle ratios to determine unknown measures and justify each result with accurate theorem-based reasoning. They will construct and label geometric figures with compass and straightedge, then use those constructions as evidence in two-column, flowchart, oral, and written proofs. Students will strengthen formal argumentation by writing and testing conjectures, converses, and contrapositives, revising weak proof steps through peer critique and proof checkpoints. They will communicate mathematical claims clearly in a courtroom-style setting by presenting a constructed figure, defended measure, and concise proof supported by precise geometric vocabulary.

Students will create a Geometry Courtroom case exhibit that includes a compass-and-straightedge construction, a solved unknown measure, and a short formal proof defending the verdict. Throughout the unit, they will build a proof portfolio with labeled constructions, corrected arguments, converse or contrapositive statements, and reflection pages that name one theorem used, one measure solved, and one revision made. They will also produce case files and proof-gallery posters that annotate complex figures with justified angle and side relationships using vertical angles, transversals, perpendicular bisectors, triangle theorems, and special right triangle ratios. By the end, each pair will submit a narrated proof board and short oral proof conference recording that explains a theorem, defends exact lengths or angle measures, and shows how feedback strengthened the final argument.

Open with a live “Geometry Courtroom” mini-trial in which you present a false geometric claim on a projected complex figure and support it with convincing but flawed reasoning about vertical angles, transversals, triangle relationships, or special right triangles. In teams, students act as jurors to inspect a companion compass-and-straightedge construction, identify one unknown measure, and prepare one justified objection using precise vocabulary, a converse or contrapositive, or a theorem-based reason. After the verdict, pairs create a quick case card that includes one claim, one solved measure, and one reason they think could hold up in court, then share it in a brief gallery walk. Close with a short proof recap in which each student names one theorem they used, one measure they found, and one step they would revise in a stronger argument.

Host a Geometry Courtroom where students present one polished case exhibit to classmates, families, or invited staff acting as jurors. Each exhibit should include a constructed and labeled figure, a solved unknown measure, a short written proof, and a brief oral defense that uses precise geometry vocabulary and responds to juror questions about theorem choice, converse or contrapositive reasoning, and revisions made after feedback. Display proof gallery posters and proof portfolios around the room so guests can examine compass-and-straightedge constructions, corrected arguments, and reflection pages before voting on whether each claim is mathematically airtight. End with a gallery walk and jury ballot that recognizes clear reasoning, accurate use of triangle and angle theorems, and strong revision across the project.