If two sides and the **contained** angle of a triangle are respectively equal to two sides and the **contained** angle of another triangle, then the triangles are congruent.
If two angles and the **contained** side of a triangle are respectively equal to two angles and the **contained** side of another triangle, then the triangles are congruent.
- Let $`k`$ be the **scale factor**, when concerning for triangle area, if the triangle area can be defined as $`\dfrac{bh}{2}`$, then by using the smaller triangles side lengths
our big triangle's area is equal to $`\dfrac{k^2bh}{2}`$. Similar equations and agruments can be dervied from this
### Side-Side-Side similarity (RRR $`\sim`$)
Three pairs of corresponding sides are in the **same ratio**
Two pairs of corresponding angles are equal. In the diagram below, we can solve for the missing angle using Angle Sum Of A Triangle Theorem (ASTT) and see that those 2 triangle's angles are equal.
|Definition|**Angle of Elevation** is the angle from the horizontal looking **up** to some object|**Angle of Depression** is the angle frorm the horizontal looking **down** to some object|
In any $`\triangle ABC`$: $`\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}`$ or $`\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}`$
We can derive the formula further to get:
- $`\dfrac{\sin A}{\sin B} = \dfrac{a}{b}`$
- $`\dfrac{\sin A}{\sin C} = \dfrac{a}{c}`$
- $`\dfrac{\sin B}{\sin C} = \dfrac{b}{c}`$
Also, for some trigonometry identities:
- $`\tan x = \dfrac{\sin x}{\cos x}`$
- $`\sin^2 A + \cos^2 A = 1`$
**If you are finding the sides or agnles of an `oblique triangle` given 1 side, its opposite angle and one other side or angle, use the sine law.**
### Ambiguous Case
The ambiguous case arises in the SSA or (ASS) case of an triangle, when you are given angle side side. The sine law calculation may need to 0, 1, or 2 solutions.
In the ambigouous case, if $`\angle A, a, b`$ are given, the height of the triangle is $`h= b\sin A`$
|Case|If $`\angle A`$ is **acute**|Condition|# & Type of triangles possible|
In any $`\triangle ABC`$, $`c^2 = a^2 + b^2 - 2ab\cose C`$
**If you are given 3 sides or 2 sides and the contained angle of an `oblique triangle`, then use the consine law**
## Directions
`Bearins`: **Always** start from **North**, and goes **clockwise**
`Direction`: Start from the first letter (N, E, S, W), and go that many degrees to the second letter (N, E, S, W)
**Note:** Northeast, Southeast, NorthWest etc. all have 45 degrees to the left or the right from their starting degree (0, 90, 180, 270)
## 2D Problems
**Note:** Watch out for the case where you don't know which side the 2 things (buildings, boats etc) are, they can result in 2 answers
## 3D problems
**Note:** Use angle theorems to find bearing/direction angle, and to help with the problem in general. Apply sine law, cosine law, and primary trigonometry ratios whenever necessary.
