Triangles

The sum of the interior angles of any triangle is always $180^\circ$.

Why it works. Imagine a triangle $ABC$. Draw a line through vertex $A$ parallel to the opposite side $BC$. This creates a straight line at vertex $A$ composed of three angles: the original angle $A$, and two angles formed by the transversal lines $AB$ and $AC$. By the alternate interior angle theorem, these two new angles are equal to angles $B$ and $C$ respectively. Since they form a straight line, $A + B + C = 180^\circ$.

SAT use. When you see a triangle with missing angles, immediately set the sum to $180^\circ$. If the triangle is part of a larger figure, look for supplementary angles outside the triangle to find the interior ones first.

Worked example. In $\triangle ABC$, $\angle A = 2x^\circ$, $\angle B = 3x^\circ$, and $\angle C = 4x^\circ$. Summing them: $9x = 180 \Rightarrow x = 20$. Thus, the angles are $40^\circ, 60^\circ, 80^\circ$.

An exterior angle of a triangle is equal to the sum of the two remote (non-adjacent) interior angles.

Why it works. Let the interior angles be $A, B, C$. The exterior angle $E$ at vertex $C$ is supplementary to $C$, so $E = 180^\circ - C$. Since $A + B + C = 180^\circ$, it follows that $A + B = 180^\circ - C$. Therefore, $E = A + B$.

SAT use. This is a massive time-saver. Instead of finding the third interior angle and then finding the supplement, jump straight to the sum of the two remote angles.

Worked example. A triangle has remote interior angles of $45^\circ$ and $72^\circ$. The exterior angle is $45^\circ + 72^\circ = 117^\circ$.

Triangles are classified by their internal angles and side lengths.

• By Angles: Acute (all $<90^\circ$), Right (one $=90^\circ$), Obtuse (one $>90^\circ$).
• By Sides: Scalene (no sides equal), Isosceles (at least two sides equal), Equilateral (all three sides equal).

SAT use. The SAT often uses these definitions to hide information. If a problem states a triangle is "isosceles," it is explicitly telling you that two angles are equal, even if the diagram doesn't mark them.

Worked example. An obtuse triangle has one angle of $110^\circ$. The other two angles must sum to $70^\circ$. If it is isosceles, the remaining angles must be $35^\circ$ and $35^\circ$.

In an isosceles triangle, the angles opposite the equal sides (base angles) are equal. Conversely, if two angles are equal, the sides opposite them are equal.

Why it works. An altitude drawn from the vertex angle to the base creates two congruent right triangles (by HL or SAS), forcing the base angles to be identical.

SAT use. Isosceles triangles are the most common source of 'hidden' information. If you see two sides marked equal, immediately mark the base angles as equal. If you see two angles equal, mark the sides as equal.

Worked example. An isosceles triangle has a vertex angle of $40^\circ$. The base angles are $(180-40)/2 = 70^\circ$.

For any triangle with sides $a, b,$ and $c$, the sum of any two sides must be strictly greater than the third: $a+b > c$, $a+c > b$, and $b+c > a$.

Why it works. The shortest distance between two points is a straight line. If $a+b = c$, the 'triangle' collapses into a line segment.

SAT use. This is tested as a range problem. If two sides are $5$ and $8$, the third side $x$ must satisfy $8-5 < x < 8+5$, so $3 < x < 13$.

Worked example. If sides are $x, 10, 10$, then $10-10 < x < 10+10$, so $0 < x < 20$.

The area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the perpendicular height to that base.

SAT use. The trap is using a side length as the height. The height must be perpendicular to the base. In a right triangle, the two legs serve as the base and height.

Worked example. A right triangle has legs of $6$ and $8$. Area $= 0.5 \times 6 \times 8 = 24$.

For a right triangle with legs $a, b$ and hypotenuse $c$, $a^2 + b^2 = c^2$.

Common Triples: $3-4-5$, $5-12-13$, $8-15-17$. Multiples of these (e.g., $6-8-10$) are also right triangles.

SAT use. Recognize triples to avoid calculation. If you see a side of $3$ and $4$, the hypotenuse is $5$. If you see $5$ and $13$, the other leg is $12$.

Worked example. A triangle has sides $10$ and $24$. $10^2 + 24^2 = 100 + 576 = 676 = 26^2$. It is a $5-12-13$ triple scaled by $2$.