Justifying Mathematical Ideas & Arguments — Module Hub & Visuals
Module 02 · 44 days · TEKS G.4 · G.5 · G.6
Lesson Slides & PDFs
Student-facing lesson slide PDFs are grouped by topic below for review, catch-up, and family support.
📖 Student Edition for this module (PDF) ✍ Skills Practice — Student Edition (PDF)
Graphic Organizers & Activities
Foldable notes and activities for each lesson, with the one to start with listed first. Sign in with your alias and PIN to open them.
Start Here — What Module 2 Is About
What this module is for, and what you will be able to do by the end of it.
Module 2 is where a drawing stops being enough. A geometric claim earns belief only when every step is justified — by a definition, a postulate, or a theorem. This lab lets you move the geometry yourself and watch the relationships that proofs are built on hold true, no matter how you drag.
Key Vocabulary
The precise words for what you just investigated. The standards this module carries are in the next section.
- Transversal — a line that crosses two or more other lines.
- Corresponding angles — same position at each intersection; equal when lines are parallel.
- Alternate interior angles — between the parallels, on opposite sides of the transversal; equal.
- Alternate exterior angles — outside the parallels, on opposite sides; equal.
- Co-interior (same-side interior) — between the parallels, same side; supplementary (sum \(180^{\circ}\)).
- Vertical angles — opposite angles at one intersection; always equal.
- Triangle Angle-Sum Theorem — the interior angles of any triangle total \(180^{\circ}\).
- Exterior Angle Theorem — an exterior angle equals the sum of the two remote interior angles.
- Kite — a quadrilateral with two pairs of consecutive congruent sides; its diagonals are perpendicular. Area \(A=\frac{d_1 d_2}{2}\), where \(d_1\) and \(d_2\) are the diagonals.
Standards in This Module
The Texas Essential Knowledge and Skills this module is built to teach. Geometry is assessed locally — there is no state EOC for this course.
G.4 logical reasoning & proof · G.5 constructions and angle/segment relationships · G.6 relationships in triangles and parallel lines.
What You're Seeing · Try This
A short tour, then four moves to make the theorems reveal themselves.
What you're seeing
In Parallel Lines & Transversal, two parallel lines (note the matching arrowheads) are crossed by one slanted line. Eight angles form. Pick a pair-family on the right and the lab shades the matching wedges and states why they're equal — or, for co-interior angles, why they add to \(180^{\circ}\).
In Triangle Angle Sum, the three interior angles update as you drag any vertex, and they always total \(180^{\circ}\). Reveal the exterior angle at \(C\) to see it equal the sum of the two remote interior angles.
Try this
- Drag the transversal handle. The four angle measures keep changing — but corresponding angles stay equal to each other. What never changes is the relationship.
- Switch to Co-interior (same-side). Read the two measures: they always sum to \(180^{\circ}\). That single fact justifies a whole step in a parallel-lines proof.
- In the triangle tool, drag a vertex until one angle is obtuse. The three still total \(180^{\circ}\) — the angle-sum theorem doesn't care what the triangle looks like.
- Turn on Exterior angle at C and drag \(A\) or \(B\). Confirm the exterior angle equals \(\angle A + \angle B\) every time.
Angle & Proof Explorer
Two live tools. Drag the handles, switch the highlighted pair, and watch the measures justify themselves.
Want to See This in 3D?
Optional, and always second. The lab above is where you measure and calculate; this is the same figure as an everyday object you can turn over and inspect. Open it when the lab already makes sense — or skip it entirely.
Where to Go Next
Back to the full course, or into the planning documents.
Instructor: Dr. Goodluck Ijezie-Desbois, PharmD · Beta Academy
Reach out by appointment, at
gijezie-desbois@betaacademy.org,
or through ParentSquare.