Triangle Similarity

Overview

Students will use similarity transformations and right-triangle angle-based side ratios to determine whether triangles are similar and to justify trig ratios for acute angles. The lesson builds from angle/side proportional reasoning toward formal similarity criteria and AA similarity.

Learning intentions

• Students will be able to use the definition of similarity to compare corresponding angles and corresponding side ratios.
• Students will be able to decide whether two triangles are similar by using similarity transformations (dilations/rigid motions) when appropriate.
• Students will be able to show why corresponding right-triangle side ratios are determined by acute angles.
• Students will be able to connect similarity to the idea behind sine and cosine for acute angles.

Success criteria

• I can identify corresponding parts and state whether angle measures match for triangles I compare.
• I can determine similarity by showing all corresponding side lengths are proportional.
• I can use similarity transformations to justify why triangles are (or are not) similar.
• I can explain why in right triangles, the side ratios for an acute angle are constant across similar triangles.

Curriculum links

• Geometry similarity transformations to decide if two figures/triangles are similar and explain similarity via equal corresponding angles and proportional corresponding sides.
• AA criterion for two triangles to be similar using properties of similarity transformations.
• Prove results about triangles using similarity, including using similarity reasoning to support right-triangle relationships.
• Understand that angle properties in right triangles lead to defining trigonometric ratios for acute angles.

Lesson structure (240 minutes)

1. 0–25 min · Retrieval warm-up + setup. Teacher displays 3 quick pairs of angle measures and side ratios (no labels yet); students compute whether proportions match and predict likely similarity, then share reasoning. Students record: “Similar likely because angles/ratios…,” and identify one question they have about correspondence.

2. 25–55 min · Mini-lesson: what similarity means. Teacher explains similarity as equality of all corresponding angles and proportionality of corresponding sides; then models finding scale factor using a dilation from one triangle to another. Students complete a short guided worksheet: mark corresponding vertices, verify angle pairs, and compute side ratios to check consistency.

3. 55–85 min · Similarity transformations (AA focus). Teacher demonstrates how a rigid motion plus dilation maps one triangle onto another; emphasizes correspondence and angle preservation. Students work in pairs on two “mapping” tasks: (a) find the scale factor, (b) verify AA by checking two angles and concluding similarity, using sentence frames: “Because ∠__ = ∠__ and ∠__ = ∠__, triangles are similar by AA.”

4. 85–115 min · Decide similarity with evidence. Teacher gives two cases: one similar pair and one non-similar pair (with tempting side proportions); students must justify “similar/not similar” using both angles and proportional sides (or transformation reasoning). Students write a 4–6 sentence justification and include a computed ratio table for sides.

5. 115–150 min · Right triangles: constant ratios from angles. Teacher leads an investigation: each group receives two right triangles with one acute angle set equal (e.g., 30° and 30°) but different sizes; students measure or use given side lengths to compute ratios like opposite/hypotenuse. Students record a claim: “If two right triangles share an acute angle, then the ratio opposite/hypotenuse matches,” and compare results across groups.

6. 150–175 min · Synthesis: why trig ratios are defined this way. Teacher connects the investigation to the idea that similar right triangles have equal corresponding side ratios; then introduces sine/cosine conceptually as ratio definitions for acute angles (no heavy computation required yet). Students complete an exit-style check: for a right triangle with acute angle θ, identify which ratio stays constant when the triangle is resized.

7. 175–220 min · Performance task: similarity justification + model. Teacher assigns a multi-part task: Students are given 2–3 triangles (some right) and must (a) determine which are similar, (b) justify using similarity transformations or AA, and (c) for any right triangles found similar, state what side ratios are equal and why. Students submit written justifications plus a labeled diagram with correspondences.

8. 220–240 min · Whole-class debrief + exit ticket. Teacher calls on groups for one “best justification.” Students complete a brief exit ticket: given two triangles, identify corresponding angles, compute one side ratio, decide similarity, and write one sentence linking similarity to constant ratios in right triangles.

Resources

• Triangle similarity guided worksheet (correspondence table + ratio checks)
• Mapping/dilation practice cards (printed triangles with grids)
• Right-triangle investigation sheet with given side lengths and/or angle values
• Graph paper or dynamic geometry tool access (optional)
• Measuring tools (protractors/rulers) if using angle/side measurement
• Sentence starters for claims and justifications
• Performance task handout with rubric/checklist
• Exit ticket slips

Assessment

• Formative: teacher circulates during similarity transformations and checks students’ correspondences and ratio tables.
• Formative: quick pair-check at 85–115 min—students must justify “similar/not similar” with at least one angle argument and one proportionality check.
• Summative for the day: performance task (diagram labels, correctness of similarity decisions, and written justification).
• Exit ticket at 220–240 min for individual accountability on AA/similarity evidence and right-triangle ratio reasoning.

Differentiation

• Support: provide sentence frames (“Triangles are similar because…,” “Correspondence is…,” “Scale factor is…”) and a correspondence key for one problem per worksheet to reduce cognitive load.
• Support: allow use of grid paper and tracing overlays for mapping; offer a bank of common ratio expressions (opposite/hypotenuse, adjacent/hypotenuse) for right-triangle tasks.
• Extension for advanced learners: ask students to create a “counterexample” pair—two triangles with proportional side lengths for only some sides but not similarity overall—and then explain exactly which angle correspondence fails.
• EAL/SEN: use structured steps and checklists; present the right-triangle investigation with numeric tables so students can focus on reasoning rather than measuring; offer reduced problem sets with the same standard-based target skills.

Extension (optional)

• Advanced extension option (if time remains during the performance task): students design their own set of three triangles where two are similar and one is not, and write a justification card for each pair including AA or transformation reasoning and a constant-ratio claim for any similar right triangles.