Home›Lesson Plans›Mathematics›Geometry · Grade 10

Similar Triangles: Proportional Shapes

Students learn that similar triangles have the same shape but different sizes — with equal angles and proportional sides — and use proportions and scale factors to find missing lengths, even measuring a tree by its shadow.

Grade 10Similarity50 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Define similar figures.
  • ✓Find a scale factor.
  • ✓Solve proportions for missing sides.
  • ✓Apply similarity to real problems.
Essential Question

How can you measure the height of a tree, a building, or a mountain without ever climbing it? Similar triangles turn a shadow into a ruler.

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

Solve with Similarity

Project each problem and have students use proportions and scale factors to find the missing length. The tool shows the setup.

📐 Use proportions — find the missing sideTry it
Use scale factor and proportions to solve each problem.
The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Measure the Unmeasurable

5 min

How can a shadow tell you a tree’s height?

👩‍🏫 Teacher Moves

  • Pose the shadow problem.
  • Ask how to solve it.
  • Set the goal: similar triangles.

🎒 Student Actions

  • Consider the problem.
  • Wonder how.
  • Predict the method.
2

I Do — Scale Factor

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Define similar (equal angles, proportional sides).
  • Find a scale factor.
  • Set up and solve a proportion.

🎒 Student Actions

  • Learn similar.
  • Find the factor.
  • Solve a proportion.
3

We Do — Solve Together

13 min

Class uses Solve with Similarity.

👩‍🏫 Teacher Moves

  • Work through each problem.
  • Set up the proportion.
  • Solve for the missing side.

🎒 Student Actions

  • Set up proportions.
  • Solve them.
  • Check the scale.
4

You Do — On Your Own

12 min

Students apply similarity.

👩‍🏫 Teacher Moves

  • Assign missing-side problems.
  • Include a real shadow problem.
  • Find scale factors.

🎒 Student Actions

  • Solve problems.
  • Do a real one.
  • Find factors.
5

Close — Close

5 min

One problem.

👩‍🏫 Teacher Moves

  • Give a similarity problem.
  • Find the missing side.
  • Hand out the exit ticket.

🎒 Student Actions

  • Solve it.
  • Show the proportion.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School: Geometry).

CCSS
G-SRT.2

Decide if two figures are similar using similarity transformations.

CCSS
G-SRT.3

Establish the AA criterion for triangle similarity.

CCSS
G-SRT.5

Use similarity criteria to solve problems and prove relationships.

CCSS
G-SRT.4

Prove theorems about triangles using similarity.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Highlight matching sides.
  • Sentence frame: “___ over ___ equals ___ over ___.”
  • Use a scale-factor chart.

Support & Access

IEP / 504
  • Start with whole-number scale factors.
  • Set up the proportion together.
  • Use a calculator.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Prove triangles similar (AA).
  • Solve indirect-measurement problems.
  • Connect similarity to trigonometry.
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻Solve with Similarity
  • 📐Rulers
  • 🧮Calculators
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

similarsame shape, proportional sizesscale factorthe ratio between similar figuresproportionan equation of two equal ratioscorresponding sidesmatching sides of similar figuresratioa comparison of two quantitiescross-multiplya method to solve proportionsAA similaritytwo equal angles prove similarityindirect measurementmeasuring using similar triangles
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
What is true about the angles and sides of similar triangles?
The angles are equal and the sides are proportional.
QUESTION 2
Two similar triangles have a scale factor of 4. A side of 3 on the small triangle matches what on the large?
12 (3 × 4).
QUESTION 3
A 5-ft person casts a 3-ft shadow while a flagpole casts a 12-ft shadow. Set up the proportion to find the pole’s height.
5/3 = x/12 (so x = 20 ft).

Going deeper? Prove it with AA.

Have students use the AA (angle-angle) criterion to prove two triangles are similar, then find missing sides. A printable similarity sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Similar TrianglesFlip
Card 1
Term
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Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Similar TrianglesQuiz
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1 / 6
Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Similar TrianglesPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: AA similarity, corresponding sides, cross-multiply, indirect measurement, proportion, ratio, scale factor, similar
  1. measuring using similar triangles
  2. a comparison of two quantities
  3. a method to solve proportions
  4. two equal angles prove similarity
  5. the ratio between similar figures
  6. same shape, proportional sizes
  7. an equation of two equal ratios
  8. matching sides of similar figures

Part B · Show what you learned

  1. What is true about the angles and sides of similar triangles?
  2. Two similar triangles have a scale factor of 4. A side of 3 on the small triangle matches what on the large?
  3. A 5-ft person casts a 3-ft shadow while a flagpole casts a 12-ft shadow. Set up the proportion to find the pole’s height.
Answer key — Part A: 1) indirect measurement · 2) ratio · 3) cross-multiply · 4) AA similarity · 5) scale factor · 6) similar · 7) proportion · 8) corresponding sides
Part B: 1) The angles are equal and the sides are proportional. 2) 12 (3 × 4). 3) 5/3 = x/12 (so x = 20 ft).