Angle-side relationships in triangles
key notes :
๐น Basic Idea
๐ In a triangle, angles and sides are related โ
๐ The longer side is opposite the larger angle, and
๐ The shorter side is opposite the smaller angle.
๐ก Rule:
โก๏ธ Bigger Angle โ Longer Opposite Side
โก๏ธ Smaller Angle โ Shorter Opposite Side
๐น Ordering Sides and Angles
๐งฉ If you know the angle measures, you can order the sides easily!
Example:
If โ A > โ B > โ C
๐ Then side BC > AC > AB
And vice versa:
If side BC > AC > AB
๐ Then โ A > โ B > โ C
๐น Triangle Inequality Theorem ๐โจ
๐ง The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
โ
a + b > c
โ
a + c > b
โ
b + c > a
๐ซ If this rule is not true, a triangle cannot be formed!
๐น Converse of the Triangle Inequality Theorem ๐
If one side of a triangle is longer, then the angle opposite that side is larger.
Example:
If side AB > side AC
๐ Then โ C > โ B
๐น Equality Case (Isosceles Triangle) ๐
In an isosceles triangle,
โจ Two sides are equal,
โจ The angles opposite those sides are also equal.
๐ Example: If AB = AC, then โ B = โ C
๐น Practical Use ๐งฎ
- 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.
๐ง Remember These Key Patterns ๐ซ
Relation | Rule |
---|---|
Larger Angle โ Longer Side | โ |
Smaller Angle โ Shorter Side | โ |
Equal Sides โ Equal Angles | โ |
a + b > c | Triangle can exist โ |
๐ Quick Example
In โณABC,
โ A = 40ยฐ, โ B = 70ยฐ, โ C = 70ยฐ
๐ Side a (BC) is shortest ๐ชถ
๐ Sides b (AC) and c (AB) are equal and longer ๐๏ธ
๐งฉ In Summary
Angles and sides in a triangle always “balance” each other!
The bigger the angle, the longer the opposite side โ
Thatโs the beauty of triangles! ๐๐บ
Learn with an example
Find the largest angle of โณVWX.

โ _____
The side lengths are VX=10 metres, VW=11 metres, and WX=13 metres. Since 10 < 11 < 13, VX < VW < WX.
Their opposite angles are in the same order, from smallest to largest:
โ W < โ X < โ V
So, the largest angle is โ V.
Find the largest angle of โณGHI.

โ ______
The side lengths are HI=35 metres, GH=43 metres, and GI=48 metres. Since 35 < 43 < 48, HI < GH < GI.
Their opposite angles are in the same order, from smallest to largest:
โ G < โ I < โ H
So, the largest angle is โ H.
Find the largest angle of โณUVW.

โ ___
The side lengths are VW=35 metres, UW=52 metres, and UV=53 metres. Since 35 < 52 < 53, VW < UW < UV.
Their opposite angles are in the same order, from smallest to largest:
โ U < โ V < โ W
So, the largest angle is โ W.

Let’s practice!