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Mathematical Architects · Geometry

Investigating Proportionality — Module Hub & Visuals

Module 03 · Similarity & Trig · TEKS G.7 / G.8

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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)

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Start Here — What Module 3 Is About

What this module is for, and what you will be able to do by the end of it.

Module 3 asks one big question: when you shrink or stretch a right triangle, what stays the same? Drag the triangle's angle below and watch two similar triangles — one big, one small — report the exact same sine, cosine, and tangent. That constancy is the whole reason right-triangle trigonometry works.

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Key Vocabulary

The precise words for what you just investigated. The standards this module carries are in the next section.

Similar figures
Same shape, possibly different size. Corresponding angles are equal and corresponding sides are proportional.
Scale factor (k)
The constant ratio between corresponding sides of two similar figures. Here, triangle 2 = \(k\times\) triangle 1.
AA similarity
If two angles of one triangle equal two angles of another, the triangles are similar. Both triangles here share \(\theta\) and a \(90^\circ\) angle.
Trigonometric ratio
For a fixed acute angle in a right triangle, the ratios opp/hyp, adj/hyp, and opp/adj are constant — that is sine, cosine, and tangent.
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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.7A G.7B G.8A G.8B

G.7 — apply similarity and AA criteria · G.8 — apply the trigonometric ratios sine, cosine, and tangent to solve right triangles.

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What You're Seeing & What to Try

A two-minute orientation, then four investigations to run yourself.

What you're seeing

  • A right triangle with the reference angle \(\theta\) at the bottom-left and the right angle at the bottom-right.
  • A second, similar triangle sharing that same angle \(\theta\), scaled by a factor \(k\) you control — same shape, different size.
  • The three trig ratios computed live: \(\sin\theta=\tfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta=\tfrac{\text{adj}}{\text{hyp}}\), \(\tan\theta=\tfrac{\text{opp}}{\text{adj}}\).
  • Side lengths for each triangle, so you can see the lengths differ while the ratios match.

Try this

  • Set \(\theta=30^\circ\) with the preset. Check that \(\sin 30^\circ = 0.5\) exactly.
  • Drag the scale factor \(k\) from \(1.2\) to \(2.4\). Watch every side length change — but the three ratios never move.
  • Drag the dot at the top of the triangle. As \(\theta\) grows, which ratio climbs fastest? (Hint: watch \(\tan\theta\).)
  • Find the angle where \(\sin\theta \approx \cos\theta\). Why does that happen near \(45^\circ\)?
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Step 5 · 2D Precision Lab

Similarity & Right-Triangle Trigonometry

Live tool — drag the dot, slide the angle, change the scale factor. The math updates in real time.

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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.

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Keep Going

Back to the course, or dig into the plan for the year.

Back to the Course

Return to the full Geometry overview — all five modules, resources, and the drafting kit.

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Course Syllabus

Policies, grading, the studio learning environment, and the full itinerary by grading period.

Read the Syllabus

Pacing Guide

Every module mapped across the grading periods — see where Module 3 falls in the year.

View the Pacing Guide