Home›Lesson Plans›Mathematics›Geometry · Grade 10

Triangle Congruence: Proving Triangles Equal

Students learn the shortcuts that prove two triangles are congruent — SSS, SAS, ASA, AAS, and HL — discovering how just a few matching parts can guarantee two triangles are identical.

Grade 10Triangles50 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…

  • ✓State the congruence criteria.
  • ✓Apply SSS, SAS, ASA, AAS.
  • ✓Know HL for right triangles.
  • ✓Know why AAA and SSA fail.
Essential Question

You don’t need to check all six parts of two triangles to know they’re identical. Which combinations of just three parts are enough to guarantee it?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Congruence Explorer · Interactive

The Congruence Explorer

Project this and tap each congruence shortcut. Have students learn what parts it requires and when it applies.

△ Explore the congruence criteria — tap eachTry it

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 — Just Three Parts

5 min

How few parts do you need to know two triangles are identical?

👩‍🏫 Teacher Moves

  • Show two triangles.
  • Ask how much you must check.
  • Set the goal: congruence criteria.

🎒 Student Actions

  • Compare triangles.
  • Wonder how few.
  • Predict the rule.
2

I Do — The Criteria

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Explain SSS and SAS.
  • Explain ASA and AAS.
  • Explain HL; note AAA/SSA fail.

🎒 Student Actions

  • Learn SSS/SAS.
  • Learn ASA/AAS.
  • Learn HL.
3

We Do — Explore Together

13 min

Class uses the Congruence Explorer.

👩‍🏫 Teacher Moves

  • Tap each criterion.
  • Identify which applies to a pair.
  • Explain the included part.

🎒 Student Actions

  • Tap each one.
  • Match a pair.
  • Explain “included.”
4

You Do — On Your Own

12 min

Students prove congruence.

👩‍🏫 Teacher Moves

  • Give triangle pairs.
  • Name the criterion that proves them.
  • Explain why some can’t be proven.

🎒 Student Actions

  • Name the criterion.
  • Justify it.
  • Spot the fails.
5

Close — Close

5 min

One pair.

👩‍🏫 Teacher Moves

  • Give two triangles.
  • Name the criterion.
  • Hand out the exit ticket.

🎒 Student Actions

  • Name the criterion.
  • Explain it.
  • 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-CO.8

Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from rigid motions.

CCSS
G-CO.7

Use the definition of congruence in terms of rigid motions.

CCSS
G-CO.6

Predict the effect of a rigid motion on a figure.

CCSS
G-SRT.5

Use congruence criteria to solve problems and prove relationships.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Criterion cards with diagrams.
  • Sentence frame: “This shows ___ because ___.”
  • Highlight the included part.

Support & Access

IEP / 504
  • Start with SSS and SAS.
  • Use marked diagrams.
  • Match criterion to a pair.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Write a two-column congruence proof.
  • Explain why SSA is ambiguous.
  • Use CPCTC.
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻The Congruence Explorer
  • 📐Rulers
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

congruentidentical in size and shapeSSSside-side-side congruenceSASside-angle-side congruenceASAangle-side-angle congruenceAASangle-angle-side congruenceHLhypotenuse-leg (right triangles)included anglethe angle between two sidescorresponding partsmatching parts of two figures
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
If two triangles have all three pairs of sides equal, which criterion proves them congruent?
SSS (side-side-side).
QUESTION 2
What does the “included” angle mean in SAS?
The angle located between the two known sides.
QUESTION 3
Do AAA (three equal angles) prove two triangles congruent? Why or why not?
No — equal angles make triangles similar (same shape) but not necessarily congruent (same size).

Going deeper? Write a proof.

Have students write a two-column proof showing two triangles are congruent, then use CPCTC (corresponding parts) to prove another fact. A printable proof 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.

🃏 Triangle CongruenceFlip
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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.

📝 Triangle CongruenceQuiz
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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.

🖨️ Triangle CongruencePrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: AAS, ASA, congruent, corresponding parts, HL, included angle, SAS, SSS
  1. the angle between two sides
  2. identical in size and shape
  3. angle-angle-side congruence
  4. side-angle-side congruence
  5. hypotenuse-leg (right triangles)
  6. matching parts of two figures
  7. side-side-side congruence
  8. angle-side-angle congruence

Part B · Show what you learned

  1. If two triangles have all three pairs of sides equal, which criterion proves them congruent?
  2. What does the “included” angle mean in SAS?
  3. Do AAA (three equal angles) prove two triangles congruent? Why or why not?
Answer key — Part A: 1) included angle · 2) congruent · 3) AAS · 4) SAS · 5) HL · 6) corresponding parts · 7) SSS · 8) ASA
Part B: 1) SSS (side-side-side). 2) The angle located between the two known sides. 3) No — equal angles make triangles similar (same shape) but not necessarily congruent (same size).