Chapter 4 Flashcards
Exterior Angle Theorem
The angle formed when you extend the side of a triangle is equal to the sum of the non-adjacent interior angles
Exterior Angle Theorem Formula
m<c’=m<a+m<b
Pythagorean theorem
a^2+b^2=c^2
SohCahToa
Sin=o/h
cos=a/h
tan=o/a
Law of Sines
a/sinA=b/sinB=c/sinC
Law of sines ambiguous case formulas
B.1= sin-1(bsinA/a)
B.2= 180-sin-1(bsinA/a)
When to use law of sines
when you have AAS, ASA, or SSA
When does the ambiguous case happen?
when the opposite side is longer than the amplitude but shorter than the other side.
b<h<a
How to solve ambiguous case?
1). find h using sohcahtoa
2). use law of sines to find first angle, using inverse function.
3). Use algebra and 180 rule to find the other angle.
The Law of Cosines (ALL 3)
a^2=b^2+c^2-2bc CosA
b^2=a^2+c^2-2ac CosB
c^2=a^2+b^2-2ab CosC
Oblique Triangle
Any triangle that is not a right triangle
Pythagorean theorem proof
1). Draw 2 squares (one inside of another)
2). Outside one has (b,a) on all 4 sides
3). inside one has “c” on all 4 sides
4) A1=(a+b)^2
5).A2= c^2
6). At=ab/2 all 4= 4(ab/2)==2ab
7). A1=A2+2ab
8). a^2+b^2+2ab=c^2+2ab
The angle bisector
An angle bisector divides the opposite side of a triangle into two segments that are proportional to the triangles other two sides.
Formula = AB/BD==AC/CD
Angle bisector proof
1). with triangle ABC extend AD to AF
2). Add a line parallel to AB connecting to C
3). Alternate interior angles BAD,DFC are congruent and DFC,CAD
4). ACF is isosceles because AC=FC
5). ADB=CDF - vertical angles
6). Triangles ADB and FDC are similar by AA. thus AB/BD==AC/CD
Geometric construction
Need only a compass and a straight edge
