**We
will prove the Law of Sines for the acute triangle case and the obtuse triangle
case.**

**Let's
go!**

**If
we want to prove the Law of Sines holds true in this acute case,
it is enough to show that.**

**Now
we have proved the Law of Sines for the Acute Triangle case!**

**Let's
continue and try to prove the Law of Sines for the Obtuse Triangle
case....**

**Let
be an obtuse triangle with . Notice again
that h is the perpendicular from the point B to the line segment
AC. H intersects the segment AC at the point D.**

**Alpha
is an obtuse angle and Beta is its supplement. Remember that angles
that are supplementary form a line.
Therefore, alpha plus Beta form the line with points A,C, and
D on it.**

**and
so h = bsin and h=asin.**

**Setting
the two h's equal to one another, we get**

**bsin=
asin.**

**Therefore,
and the Law of Sines works for the Obtuse Triangle
case as well:)**

**We
will prove the Law
of Cosines
for the acute
triangle case
and the obtuse
triangle case.**

**Let's
go!**

**Remember
the cosine of an angle is equal to the .**

**Segment
CD or h is the perpendicular from the point C to the line segment
AB. It intersects the line segment AB at the point D. **

**.
Therefore,
.**

**Therefore,
we have proved the Law of Cosines for the Acute Triangle case.**

**Using
the fact that Triangle ADB and Triangle CDB are both right triangles,
we can use the Pythagorean Theorem to get...**

**Then
finally, . So, we have proved the Law of Cosines for the Obtuse
Triangle Case.**