Unit 4
Triangle Theorems
Unit 4 takes the proof architecture students built in Unit 3 and puts it to work on triangle geometry. Students start with a familiar “fact” — that triangle angles sum to 180° — and experience it first as a conditional proof whose missing links are honest and visible. Across the unit, they close that chain by proving the constructions it depends on, then extend the same flow‑chart and paragraph proof habits to the base angles theorem, the bisector theorems, and polygon angle sums. By the end, students see triangle theorems not as rules to memorize, but as named tools in a deductive system they have helped construct.
For the full story of how this unit is sequenced, what changed from Year 1, read the complete Unit 4 overview and navigation instrument below.
Unit 4 Lessons
Lesson 4.1 – Triangle Angle Sum, Exterior Angles, and Polygon Sums
Students revisit a “known” fact — that triangle angles sum to 180° — and discover it as a conditional proof with visible gaps. Along the way they connect triangle angle sum, exterior angles, and polygon sums, while naming exactly which unproven constructions their argument depends on.
Lesson 4.2 – Three Constructions and Their Proofs
Students prove three core constructions: copying an angle, building a parallel through a point, and dropping an altitude. Each construction is justified with SSS, SAS, and CPCTC, closing the logical chain that turns the conditional proof from Lesson 4.1 into a fully established theorem.
Lesson 4.3 – The Base Angles Theorem and Its Converse
Starting from a two‑hikers scenario, students uncover and prove the base angles theorem using the angle bisector as a tool. They then confront why the converse is a separate claim, meeting their first major biconditional that ties equal sides and equal angles together.
Lesson 4.4 – The Bisector Theorems
Through beacon and water‑tower contexts, students discover the angle bisector and perpendicular bisector theorems. They formalize “equidistant” and “perpendicular distance,” then use triangle congruence to prove that bisectors are sets of points satisfying precise distance conditions.
Unit 4 Assessment
I’m happy to share my Unit 4 assessments and grading rubrics for teacher use, but I don’t want them publicly available on the internet. If you’d like a copy, please email me from your school account at jay@thinkingthroughgeometry.net and let me know which school and state you teach in.