Triangles pop up all over the Math section. There are questions specifically about triangles, questions that ask about triangles inscribed in polygons and circles, and questions about triangles in coordinate geometry.
Three Sides, Four Fundamental Properties
Every triangle, no matter how special, follows four main rules.
1. Sum of the Interior Angles
If you were trapped on a deserted island with tons of SAT questions about triangles, this is the one rule you’d need to know:
The sum of the interior angles of a triangle is 180°.
If you know the measures of two of a triangle’s angles, you’ll always be able to find the third by subtracting the sum of the first two from 180.
2. Measure of an Exterior Angle
The exterior angle of a triangle is always supplementary to the interior angle with which it shares a vertex and equal to the sum of the measures of the remote interior angles. An exterior angle of a triangle is the angle formed by extending one of the sides of the triangle past a vertex. In the image below, d is the exterior angle.
Since d and c together form a straight angle, they are supplementary: 
. According to the first rule of triangles, the three angles of a triangle always add up to 
, so 
. Since 
and 
, d must equal a + b.
3. Triangle Inequality Rule
If triangles got together to write a declaration of independence, they’d have a tough time, since one of their defining rules would be this:
The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
There you have it: Triangles are unequal by definition.
Take a look at the figure below:
The triangle inequality rule says that c – b < a < c + b. The exact length of side a depends on the measure of the angle created by sides b and c. Witness this triangle:
Using the triangle inequality rule, you can tell that 9 – 4 < x < 9 + 4, or 5 < x < 13. The exact value of x depends on the measure of the angle opposite side x. If this angle is large (close to 
) then x will be large (close to 13). If this angle is small (close to 
), then x will be small (close to 5).
The triangle inequality rule means that if you know the length of two sides of any triangle, you will always know the range of possible side lengths for the third side. On some SAT triangle questions, that’s all you’ll need.
4. Proportionality of Triangles
Here’s the final fundamental triangle property. This one explains the relationships between the angles of a triangle and the lengths of the triangle’s sides.
In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.
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