Ordering Sides by Angle Size
In any triangle, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. Angles and sides are in the same order.
Introduction
This is the “opposite relationship”: to find the longest side, find the largest angle and look across from it. To find the smallest angle, find the shortest side and look at the angle across from it. This ordering works both ways.
Past Knowledge
Triangle Inequality (6.2.1). Angle Sum Theorem (5.1.2). Isosceles properties (5.1.4).
Today's Goal
Order sides from angles, and order angles from sides.
Future Success
Hinge Theorem (6.2.3), indirect proofs, optimization.
Key Concepts
Angle-Side Relationship
From Angles → Sides: If , then (side is opposite )
From Sides → Angles: If , then
Key Fact: “Opposite” Means Across
Side (opposite ) connects the other two vertices: and . The angle and its opposite side never share a vertex.
Worked Examples
Ordering Sides from Angles
In , , , . Order the sides from shortest to longest.
Smallest angle: → shortest side: (opposite )
Largest angle: → longest side: (opposite )
Ordering Angles from Sides
In , . Order the angles from largest to smallest.
Longest side: → largest angle: (opposite )
Shortest side: → smallest angle: (opposite )
Combined Reasoning
In , and . Is longer or shorter than 10?
is opposite .
Since , the other angles sum to 120°.
If , then . If , then .
It depends on — we need more info. (This teaches that one angle alone isn't enough.)
Common Pitfalls
Matching Angle to Adjacent Side
The largest angle corresponds to the opposite side, NOT a side that touches it. 's opposite side is , not or .
Thinking the Relationship Is Addition
The theorem says the order of angles matches the order of sides. It does not say the angle measure equals or is proportional to the side length.
Real-Life Applications
Rock Climbing — Angle of Approach
Climbers choose routes based on angles. A steeper wall (larger angle at the base) means a longer route to the top. The angle-side relationship helps estimate distances without direct measurement.
Landscaping — Sloped Terrain
When designing drainage on triangular lots, landscapers use this relationship: the side with the steepest slope (opposite the widest angle) needs the most drainage infrastructure.
Practice Quiz
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