Triangle Properties on the SAT
Angles and Side Relationships
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Triangle Properties
Triangles appear everywhere on the SAT because a few reliable properties solve a wide range of questions. If you know the angle sum, the triangle inequality, and the longest side rule, you can move quickly.
This lesson focuses on those core properties and shows how to turn a diagram into a simple equation. We will also clarify when to use each rule so you do not mix up side length and angle relationships.
A Simple Definition Unlocks Triangle Properties
Every triangle has an angle sum of $180^\circ$. If two angles are known, the third is just what is left. This rule alone solves many SAT triangle questions.
The triangle inequality says the sum of any two sides must be greater than the third. Meanwhile, the largest angle is opposite the longest side. These facts help you compare lengths and identify possible triangles.
Work Through Triangle Properties Step by Step
A triangle has angles $50^\circ$, $60^\circ$, and $x$. What is $x$?
Use the angle sum rule to find a missing angle and confirm it fits the triangle.
Use the fact that the triangle angle sum is $180^\circ$.
Solve for $x$ to isolate the variable.
Use Desmos to Check Triangle Properties
Do side lengths $a$, $b$, and $c$ satisfy the triangle inequality $a + b > c$?
Desmos is rarely faster for basic triangle properties, but it can help check if side lengths form a valid triangle by comparing sums.
7 + 8 > 12
Most of the time, mental math and one step algebra is faster here.
Desmos can help with arithmetic checks, but geometry rules are faster and more reliable for most triangle property questions.
Expert move: Use Desmos only to add the two smaller sides and compare to the largest side; the triangle-inequality logic is still the key step.
When to skip Desmos: If the numbers are simple, do the check mentally and move on.
- Desmos features used: quick numeric checks.
- Common mistake: forgetting that the largest angle is opposite the longest side.
Practice Triangle Properties with SAT-Style Questions
Apply triangle rules to solve each question.
A triangle has angles and . What is the third angle?
Which set of side lengths can form a triangle?
In a triangle, the longest side is opposite which angle?
Two sides of a triangle are and . Which value could be the third side?
Key Takeaways to Remember for Triangle Properties
- Triangle angles sum to $180^\circ$.
- The longest side is opposite the largest angle.
- The sum of any two sides must exceed the third.
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