Lesson 3: Triangle Properties and Congruence

👨🏽‍🏫 Introduction

Have you ever folded a paper triangle to match it perfectly with another one? Or noticed how triangular road signs are always balanced? That’s the power of triangle properties and congruence in action! 🔺

In this lesson, you’ll learn how to identify different triangles, apply angle rules, and prove when two triangles are exactly alike using congruence rules.

• 📐 Understand types of triangles by sides and angles
• 🎯 Learn triangle angle properties (interior and exterior)
• 🔁 Master congruence rules (SSS, SAS, ASA, RHS)

🧠 Key Concepts

📌 1. Triangle Basics

• Sum of interior angles in a triangle is always
• Exterior angle = sum of the two opposite interior angles

📐 2. Types of Triangles by Sides

• Equilateral: All sides and angles are equal
• Isosceles: Two sides and angles are equal
• Scalene: All sides and angles are different

Equilateral
Isosceles
Scalene

📐 3. Types of Triangles by Angles

• Acute: All angles < 90°
• Right: One angle = 90°
• Obtuse: One angle > 90°

Acute
Right
Obtuse

📌 4. Triangle Congruence Rules (with Improved Diagrams)

Two triangles are congruent if they have the same shape and size — even if flipped or rotated.

• SSS: All three sides equal
• SAS: Two sides and the included angle equal
• ASA: Two angles and the included side equal
• RHS: Right angle, hypotenuse, and one other side equal

📐 SSS (Side-Side-Side)

📐 SAS (Side-Angle-Side)

📐 ASA (Angle-Side-Angle)

📐 RHS (Right-Angle-Hypotenuse-Side)

🔍 Worked Examples

1. 1. Angle Sum: A triangle has angles 50° and 60°. Find the third angle.
      

1. 1. Exterior Angle: One exterior angle is 120°, and one opposite interior angle is 50°. Find the other.
      

1. 1. Congruence – SAS: AB = DE = 5 cm, AC = DF = 7 cm, → Congruent by SAS ✅

1. Congruence – RHS: Right-angled triangles with equal hypotenuse (10 cm) and one side (6 cm) → Congruent by RHS ✅

✍🏽 Practice Exercises

🅰️ Section A – Triangle Angles

1. Find the missing angle in a triangle with angles 40° and 85°.
2. If an exterior angle is 110° and one opposite interior angle is 45°, find the other.

🅱️ Section B – Triangle Classification

1. Classify a triangle with sides 6 cm, 6 cm, and 8 cm.
2. What type of triangle has angles 40°, 60°, and 80°?

🅾️ Section C – Triangle Congruence

1. Which rule proves congruence for two triangles with all sides equal?
2. Two triangles are right-angled, with equal hypotenuse and one equal side. What rule applies?

✅ Answers and Explanations

• A1:
• A2:
• B1: Isosceles (two equal sides)
• B2: Acute triangle (all angles < 90°)
• C1: SSS
• C2: RHS

📌 Recap

• 🔺 Triangle angles sum up to 180°
• 📐 Exterior angle = sum of two opposite interior angles
• ✅ Use SSS, SAS, ASA, or RHS to prove congruence

💭 Reflection Prompt

Draw two triangles that look different. Are they congruent? Can you prove it using one of the rules (SSS, SAS, ASA, RHS)? Which rule fits best?