Question

GRE Review: Triangle Geometry

Answer 1

\({\bf{Memorization~List:}}\)
- all triangles have angles that sum up to 180
- the exterior angle on a triangle is equal to the sum of the two non-adjacent angles



so in the above example, angles 4 and 5 would both be equal to (angle 1 + angle 2)

if you don't remember this fact you can always verify manually

angle 1 + angle 2 + angle 3 = 180 since they're three angles in a triangle
angle 3 + angle 4 = 180

subtracting angle 3 from both equations gives us

angle 1 + angle 2 = 180 - angle 3
angle 4 = 180 - angle 3

so angle 4 = angle 1 + angle 2
same logic extends to angle 5

Answer 2

- side length rules: let a, b, and c be the lengths of a triangle
> a^2 + b^2 = c^2 gives a right triangle. note that this is the pythagorean theorem
> a^2 + b^2 < c^2 gives an acute triangle
> a^2 + b^2 > c^2 gives an obtuse triangle

> the sum of any two sides must be greater than the third side, or in other words
a + b > c
a + c > b
b + c > a
if any of these conditions fails then the triangle is not valid and could not actually exist

> the difference of the lengths must be less than the third side, or in other words
a < |c - b|
b < |a - c|
c < |b - a|

Answer 3

special triangles: memorize these angles and proportions to save yourself time w/ calculations



these can be derived w/ pythagorean theorem in a pinch

special formulas:

area of triangle = (1/2)bh, make sure the height is drawn perpendicularly from the base otherwise it won't work

area of an equilateral triangle: you can calculate the height using the pythagorean theorem but as a shortcut, if you know the side length, it's A = (sqrt(3)/4 ) s^2

Answer 4

Adapted from Barron's test prep book for the new GRE, 19th edition