Criteria for Congruency of Triangles (I)

Content Standards

Students will understand and demonstrate the criteria of congruency of triangles, including the concept of superimposition and the specific conditions under which two triangles are congruent.

Performance Standards

Students will be able to:

Identify and define congruent triangles.
Explain different congruency criteria (SSS, SAS, ASA, RHS).
Apply appropriate congruency criteria to prove two triangles are congruent.
Solve geometrical problems using triangle congruency.

Alignment Standards

Reference: NCERT Book Alignment

The lesson is aligned with the NCERT Grade 9 Maths Book-Chapter 7: Triangle, Section: 3 “Criteria for Congruence of Triangles (I)”

Learning Objectives

By the end of this lesson, students will be able to:

Understand the meaning of congruence in triangles.
Identify and apply different criteria of congruence — SSS, SAS, ASA, AAS, and RHS.
Demonstrate congruence using physical and virtual models.
Relate congruence to real-life examples and geometric proofs.

Prerequisites (Prior Knowledge)

Students should already:

Understand basic properties of triangles.
Know the concept of equal sides and angles.
Recall geometric terms like vertices, base, altitude, and angle.

Introduction

Students often notice shapes that look “exactly the same” — like tiles, traffic signs, or window panes. In this session, they’ll learn what makes two triangles exactly identical in shape and size, and how mathematicians prove this using congruency criteria.

Timeline (40 Minutes)

Title	Approximate Duration	Procedure	Reference Material
Engage	5	Show Two paper triangles placed side by side — one exactly overlaps the other when superimposed. And Ask: Are these triangles identical? How can, we be sure? How can we prove that two triangles are exactly the same without measuring every side and angle? 2. Show two triangles -one matching and one slightly larger. Discuss what makes them look same or different. Purpose: Leads students to discover that specific criteria can confirm congruence without measuring everything.	Slides + real world object
Explore	10	Teacher Introduction: Today we’ll explore how we can decide when two triangles are exactly the same — not just similar in look, but identical in size and shape. Let’s think about some real situations.” A triangle is drawn on paper, and its mirror image is taken in a vertical mirror. Are the two triangles congruent? You have two triangular metal plates. You measure all three sides of each triangle and find them equal. Do you think the triangles are identical in shape? Expected Student Response: “They look the same in size but flipped or reversed.” “Yes, they will be the same shape.”	Slides
Explain	10	*Teacher Explanation:* Introduce the term Congruence — meaning exactly the same shape and size. Explain each criterion with diagrams: 1. *SSS* (Side-Side-Side) 2. *SAS* (Side-Angle-Side) 3. *ASA* (Angle-Side-Angle) 4.AAS (Angle-Angle-side) 5.RHS (Right-Hypotenuse-Side) Show examples and non-examples using visuals. Ask conceptual questions: “If two sides and the included angle are equal, can triangles differ?”	Slides and Virtual Lab
Evaluate	10	Students will attempt the Self Evaluation task on LMS.	Virtual Lab
Extend	5	Scenario Thinking: Why is AAA not a criterion for congruency though it shows similarity between triangles?	Slides

Criteria for Congruency of Triangles (I)

Introduction

In this lesson, students will explore the conditions under which two triangles are congruent—that is, exactly the same in shape and size. Using real-life examples, virtual models, and interactive geometry tools, they will understand how sides and angles determine congruence and how these properties are used in problem-solving and geometric proofs.

Theory

Introduction: Why Learn About Congruent Triangles?

Have you ever noticed how two identical window panes fit perfectly in a frame, or how two blades of a windmill look exactly alike?

That’s congruence in geometry — when two shapes are identical in all respects.

Understanding congruence helps us design accurate objects, build symmetrical structures, and reason logically in geometry.

What Are Congruent Triangles?

Two triangles are said to be congruent if all three sides and all three angles of one triangle are equal to the corresponding sides and angles of the other triangle.

However, we don’t need to check all six measures every time — mathematicians have found certain rules or criteria that are enough to prove congruence.

Steps / Process / Rules

Criteria for Congruence of Triangles

There are several conditions (criteria) by which two triangles can be proven congruent without checking all sides and angles.

Side-Side-Side (SSS) Criterion
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
Example: If AB = DF, BC = FE, and CA = ED, then △ABC ≅ △DFE.
Side-Angle-Side (SAS) Criterion
If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, they are congruent.
Example: If AB = PQ, ∠B = ∠Q, and BC = QR, then △ABC ≅ △PQR.
Angle-Side-Angle (ASA) Criterion
If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, the triangles are congruent.
Example: If ∠B = ∠E, BC = EF, and ∠C = ∠F, then △ABC ≅ △DEF.
Angle-Angle-Side (AAS) Criterion
If two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.
Example: If ∠A = ∠P, ∠B = ∠Q, and BC = QR, then △ABC ≅ △PQR.
RHS (Right angle–Hypotenuse–Side) Criterion:
This criterion applies only to right-angled triangles. If the hypotenuse and one side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the triangles are congruent. Example: If △ABC and △DEF are right-angled at B and Q respectively, and hypotenuse AC = PR and side AB = PQ, then △ABC ≅ △PQR.

Visual Representation

Paper Folding Experiment:
Cut out two triangles from paper with the same side lengths. When you place one on top of the other, they fit exactly — showing SSS congruence.
Interactive Geometry App (GeoGebra / VR Model):
Students can adjust side lengths and angles to observe when triangles remain congruent or not.
Real-Life Example:
Bridge trusses, iron frames, or roof supports use congruent triangles to ensure balance and equal force distribution.

Applications / Why is it Useful?

Congruence is used in many real-life and mathematical situations:

Architecture: Ensuring walls, frames, and beams are identical for stability.
Engineering: Designing parts that must fit together perfectly.
Art and Design: Creating patterns or tiling where shapes must match exactly.
Mathematics: To prove other geometric properties such as isosceles triangle theorems or Pythagoras-based constructions.

Vocabulary

This is the list of vocabulary terms used throughout the lesson.

Congruent Figures:Figures that have the same shape and size..
Congruent Triangles:Triangles with equal corresponding sides and angles..
SSS Criterion: Side–Side–Side rule for proving triangles congruent.
SAS Criterion: Side–Angle–Side rule for proving triangles congruent.
ASA Criterion: Angle–Side–Angle rule for proving triangles congruent.
AAS Criterion: Angle–Angle–Side rule for proving triangles congruent.
Included Angle: The angle formed between two given sides.
Included Side: The side that lies between two given angles.
Corresponding Parts: Matching sides or angles in two triangles.
RHS Criterion: Right angle–Hypotenuse–Side rule used for right-angled triangles.
Hypotenuse: The longest side in a right-angled triangle, opposite the right angle.

Criteria for Congruency of Triangles (I)

Introduction

Welcome to the Triangle Congruence VR Lab, an immersive 3D learning experience where you’ll explore how two triangles can be proven congruent using different criteria.
This guide will walk you through each stage of the virtual activity — from introduction to interactive learning and final evaluation.

Key Features

A 3D virtual stage displaying two triangles for comparison.
Interactive info panels and voice narration explaining each congruence rule.
Click-based interaction: students attach missing sides or angles to explore each condition of congruence.
Real-time animations showing triangles overlapping to visually confirm congruence.
A final in-VR quiz to test understanding through short questions.

Step-by-Step Procedure for VR Experience

Step 1: Enter the Virtual Lab

Students enter a 3D environment where an information panel appears on the screen.
The message reads: “Welcome! Today, you will learn about the Criteria of Congruence of Triangles.”
Two triangles appear on the stage as a visual introduction.
Click Next to begin the exploration.

Step 2: Understanding SAS (Side–Angle–Side) Congruence

The narrator and info panel explain that if two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
Animation shows the two triangles gradually overlapping to demonstrate equality.
Students are prompted to click on one side of the second triangle (not yet connected).
Once clicked, the missing side attaches to form the SAS condition, showing both triangles as congruent.

Step 3: Exploring Other Criteria (SSS, ASA, RHS, AAS)

Each criterion is shown one by one in the same interactive style:

SSS (Side–Side–Side): Three sides of both triangles appear. Students click to align them; triangles overlap completely.
ASA (Angle–Side–Angle): Two angles and the included side are highlighted; students click to join and see the overlap.
RHS (Right angle–Hypotenuse–Side): Right-angled triangles appear; students click to connect the hypotenuse and side, confirming congruence.
AAS (Angle-Side-Angle): The two angles and one excluded side are highlighted.

Step 4: Evaluation

After interaction, students proceed to the quiz:
- 2 MCQs