The Hinge Theorem (SAS Inequality)
If two triangles have two pairs of sides congruent, but the included angles differ, then the triangle with the larger included angle has the longer third side — like opening a door wider.
Introduction
Think of a door hinge: two arms of fixed length are connected at a pivot. As you open the door wider (increase the angle), the gap between the door's edge and the frame (the third side) gets larger. That's exactly the Hinge Theorem.
Past Knowledge
SAS (5.2.3). Triangle Inequality (6.2.1). Angle-side ordering (6.2.2).
Today's Goal
State and apply the Hinge Theorem and its converse.
Future Success
Similarity (Ch 7), indirect proofs, optimization problems.
Key Concepts
Hinge Theorem (SAS Inequality)
If two sides of one triangle are congruent to two sides of another, and the included angle of the first is larger, then the third side of the first is longer.
Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another, and the third side of the first is longer, then the included angle of the first is larger.
Formal Statement
If , , and , then .
Theorem & Proof
Two-Column Proof: Hinge Theorem
Given: , ,
Prove:
Strategy: Position inside using rigid motions, then apply the Triangle Inequality.
Same two sides, different included angles → the wider angle opens a longer opposite side.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Map so and aligns with . Then maps to (since ). | Rigid motion (translation + rotation preserving lengths) |
| 3 | maps to a point on with (so may not coincide with ) | , but so lands inside |
| 4 | Draw . Since is between and , point is inside angle | , so the image of falls strictly inside |
| 5 | Rigid motions preserve distances | |
| 6 | In : is true, but we need the other direction. Bisect to locate relative to | Triangle Inequality in |
| 7 | , so | The angle bisector of creates isosceles sub-triangles showing ; combined with step 5, |
∎ Wider angle → longer opposite side when two sides are fixed.
Worked Examples
Direct Hinge Comparison
: . : . Compare and .
Same two sides ().
(Hinge Theorem: larger angle → longer third side)
Converse — Comparing Angles
: . : . Compare and .
Same two sides ().
(Converse of Hinge Theorem: longer third side → larger included angle)
Finding an Inequality for x
Two triangles share sides 9 and 12. In the first, the included angle is and the third side is . In the second, the included angle is and the third side is . Set up and solve the inequality.
→ by Hinge Theorem:
Also: third sides must be positive → and . Combined:
Common Pitfalls
Comparing Triangles Without Equal Sides
The Hinge Theorem requires two pairs of sides to be congruent. If the sides are different, you cannot apply it — use different reasoning.
Reversing the Direction
Larger angle → longer side. Don't confuse the Hinge Theorem direction with its converse (longer side → larger angle). Both are true, but make sure you apply the right one.
Real-Life Applications
Robotics — Arm Reach
A robot arm with two fixed-length segments pivoting at a joint follows the Hinge Theorem exactly. Opening the joint angle increases the distance from the base to the end-effector — essential for calculating workspace boundaries.
Biomechanics — Jaw Opening
Your jaw is a hinge. The mandible and skull form two fixed “sides.” As you open your mouth wider (increasing the hinge angle), the gap between your teeth (the third side) increases. Dentists use this relationship for bite analysis.
Practice Quiz
Loading...