Angle-side relationships in triangles

key notes :

🔹 Basic Idea

👉 In a triangle, angles and sides are related —
📏 The longer side is opposite the larger angle, and
📐 The shorter side is opposite the smaller angle.

💡 Rule:

➡️ Bigger Angle → Longer Opposite Side
➡️ Smaller Angle → Shorter Opposite Side

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🔹 Ordering Sides and Angles

🧩 If you know the angle measures, you can order the sides easily!
Example:
If ∠A > ∠B > ∠C
👉 Then side BC > AC > AB

And vice versa:
If side BC > AC > AB
👉 Then ∠A > ∠B > ∠C

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🔹 Triangle Inequality Theorem 📐✨

🧠 The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

✅ a + b > c
✅ a + c > b
✅ b + c > a

🚫 If this rule is not true, a triangle cannot be formed!

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🔹 Converse of the Triangle Inequality Theorem 🔄

If one side of a triangle is longer, then the angle opposite that side is larger.
Example:
If side AB > side AC
👉 Then ∠C > ∠B

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🔹 Equality Case (Isosceles Triangle) 💖

In an isosceles triangle,
✨ Two sides are equal,
✨ The angles opposite those sides are also equal.

📏 Example: If AB = AC, then ∠B = ∠C

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🔹 Practical Use 🧮

• Used in geometry proofs and construction problems.
• Helps in ranking sides or angles by size.
• Important in real-life applications like design, construction, and navigation.

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🧠 Remember These Key Patterns 💫

Relation	Rule
Larger Angle ↔ Longer Side	✅
Smaller Angle ↔ Shorter Side	✅
Equal Sides ↔ Equal Angles	✅
a + b > c	Triangle can exist ✅

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🌟 Quick Example

In △ABC,
∠A = 40°, ∠B = 70°, ∠C = 70°
👉 Side a (BC) is shortest 🪶
👉 Sides b (AC) and c (AB) are equal and longer 🏗️

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🧩 In Summary

Angles and sides in a triangle always “balance” each other!
The bigger the angle, the longer the opposite side —
That’s the beauty of triangles! 💖🔺

Learn with an example

Let’s practice!