Geometry Cheat Sheet: Triangles

Posted on January 7, 2013 by Anthony Ritz

In each Cheat Sheet, I’ll cover, as succinctly as possible, every rule you absolutely must know to solve problems in a single area found on standardized tests.

Learning these rules isn’t a substitute for developing higher-order problem-solving and strategic thinking skills; rather, it’s a necessary precondition and foundation for all of that strategizing to take place. This Cheat Sheet lists the minimum requirements to get your foot in the door. It’s the price of admission.

If you’re taking any test involving geometry, here’s what you need to know about triangles:

Table of Contents

Lines and Angles
Triangles
Quadrilaterals
Other Polygons
Circles
3-D Geometry
Coordinate Geometry

Triangle Basics

Area $=\frac{1}{2}bh$
Total angle measure $=180^\circ$

The Pythagorean Theorem:

For a right triangle with side lengths $a$ , $b$ , and $c$ ,
$a^2+b^2=c^2$ , where $c$ is the hypotenuse (the longest side).

The Triangle Inequality:

For any with side lengths $a$ , $b$ , and $c$ ,
$a+b>c$ , $a+c>b$ , and $b+c>a$ .
(The sum of the lengths of any two sides must exceed the length of the third side.)

This is not a triangle.

Special Triangles

Common Pythagorean Triples

Any multiple of a Pythagorean triple is also a Pythagorean triple.

$3-4-5$ / $6-8-10$ / $9-12-15$ / $etc.$
$5-12-13$ / $10-24-26$ / $etc.$

The $45^\circ-45^\circ-90^\circ,$ $x-x-x\sqrt{2}$ Isosceles Right Triangle

The Equilateral Triangle and the $30^\circ-60^\circ-90^\circ,$ $x-x\sqrt{3}-2x$ Right Triangle

An equilateral triangle has three sides of equal length and three angles of equal $60^\circ$ measure.

Area of an equilateral triangle (with side length $s$ ) $=(\frac{s}{2})^2\sqrt{3}$ .

Isosceles Triangles

An isosceles triangle has at least two sides of equal length.
The angles opposite the equal sides of an isosceles triangle are equal.

$AC=BC$

Triangle Similarity

Similarity Basics

Two objects are similar if they have the same shape, aside from scaling. In other words, similarity means that either object can be scaled up or down (and possibly flipped) to be identical the other.
If two triangles are similar, then all of their corresponding angles are equal and all of their corresponding lengths scale by a common scaling factor, $\phi$ . Furthermore, the triangles’ areas scale by the factor $\phi^2$ .
Note: Every geometric figure has similar versions, but triangles are the most-tested shapes on similarity questions.

Establishing Similarity in Triangles

Similarity in triangles may be established using Angle-Angle-Angle, Side-Side-Side, or Side-Angle-Side comparison.
Note: Similarity is most commonly tested on standardized tests using Angle-Angle-Angle similarity.

Angle-Angle-Angle

The two triangles are similar if $\alpha_1=\alpha_2$ , $\beta_1=\beta_2$ , and $\gamma_1=\gamma_1$ .

Side-Side-Side

The two triangles are similar if $\frac{a}{A}=\frac{b}{B}=\frac{c}{C}$ .

Side-Angle-Side
The two triangles are similar if $\alpha_1=\alpha_2$ and $\frac{a}{A}=\frac{b}{B}$ .
Note: Side-Angle-Side similarity requires that the equal angles must be located between the pair of corresponding sides.