Triangles are the most-tested geometric shapes on the Digital SAT. Understanding their fundamental properties — angle sums, exterior angles, types, and inequalities — provides the foundation for more complex geometry problems.
Core Concepts
Triangle Angle Sum Theorem
The three interior angles of any triangle sum to :
Exterior Angle Theorem
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles:
Types of Triangles by Angles
- Acute: all angles < 90°
- Right: one angle = 90°
- Obtuse: one angle > 90°
Types of Triangles by Sides
- Equilateral: all sides equal, all angles = 60°
- Isosceles: two sides equal, base angles equal
- Scalene: no sides equal
Triangle Inequality Theorem
The sum of any two sides must be greater than the third side:
, ,
Relationship Between Sides and Angles
The longest side is opposite the largest angle. The shortest side is opposite the smallest angle.
Strategy Tips
Tip 1: Use to Find Missing Angles
If you know two angles, the third is .
Tip 2: Exterior Angle Shortcut
The exterior angle theorem saves a step compared to finding the third interior angle first.
Tip 3: Isosceles Triangle = Two Equal Angles
If two sides are equal, the angles opposite them are equal.
Worked Example: Example 1
A triangle has angles and . Find the third angle.
Worked Example: Example 2
An exterior angle is . The two remote interior angles are and . Find .
→ →
Worked Example: Example 3
Can a triangle have sides 3, 5, and 9?
. No — violates the triangle inequality.
Worked Example: SAT-Style
An isosceles triangle has a vertex angle of . Find the base angles.
Base angles: each.
Practice Problems
Problem 1
Angles are , , and . Find all three angles.
Problem 2
Can a triangle have sides 7, 10, and 18?
Problem 3
An exterior angle of a triangle is . One interior angle is . Find the other interior angle adjacent to the exterior angle.
Problem 4
An equilateral triangle has perimeter 24. Find the side length.
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Common Mistakes
- Forgetting the angle sum is 180°, not 360°. 360° is for quadrilaterals.
- Misapplying the exterior angle theorem. The exterior angle equals the sum of the TWO remote (non-adjacent) interior angles.
- Triangle inequality. Check all three conditions, not just one.
Key Takeaways
Interior angles sum to .
Exterior angle = sum of two remote interior angles.
Isosceles: two equal sides → two equal angles.
Equilateral: all 60° angles.
Triangle inequality: any two sides must sum to more than the third.
Largest angle is opposite the longest side.