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Think back to geometry. Write down the ways to prove that two triangles are congruent.
Warm-Up: May 20, 2013
The Law of SinesSection 6.1
For a triangle with angle measures A, B, C and side lengths opposite those angles of a, b, c:
True for any triangle (acute, right, obtuse)
The Law of Sines
C
c
B
b
A
a
sinsinsin
c
C
b
B
a
A sinsinsin
Solving a triangle means finding all side lengths and angle measures
Use Law of Sines A+B+C=180˚ For examples, you-try’s and homework,
round side lengths and angle measures to 3 decimal places.
Law of Sines can be used to solve a triangle if 2 angles and 1 side are known (SAA or ASA) or two sides and an angle opposite one of them (SSA)
Solving a Triangle
1. Start by drawing a rough sketch of a triangle (not to scale)
2. Find the third angle measure (A+B+C=180˚)
3. Use Law of Sines twice to find the two missing side lengths
4. Draw a better version of your triangle (to scale)
5. Check that your longest side is across from biggest angle, shortest side is across from smallest angle
Solving a Triangle: SAA or ASA
Solve the triangle
Example 1: SAA
24
56
22
C
B
A
c
b
a
Solve the triangle
You-Try #1: SAA
82
64
14
C
B
A
c
b
a
Solve the triangle
Example 2: ASA
C
B
A
c
b
a
35
85
30
Solve the triangle
You-Try #2: ASA
5.22
40
12
C
B
A
c
b
a
Page 607 #1-13 Every Other Odd Page 608 #17-29 Every Other Odd Page 608 #33-45 Every Other Odd
Assignments
Find all angles θ in the interval [0, 2π) such that
Warm-Up: May 21, 2013
2
1sin
Homework Questions?
SSA: The Ambiguous Case Given two side lengths and an angle
opposite one of them, there could be 0, 1, or 2 triangles
Assume a, b, and A are given If , then there are no triangles If , then there is one right triangle If , then there are two
triangles If , then there is one triangle
The Law of Sines will give you the number of triangles
SSA: How Many Triangles?
Aba sinAba sin
baAb sinba
Solve the triangle
Example 3: SSA
C
B
A
c
b
a
20
40
30
Solve the triangle
Example 4: SSA
C
B
A
c
b
a
50
20
30
Solve the triangle
Example 5: SSA
C
B
A
c
b
a
150
30
10
Solve the triangle
You-Try #5: SSA
C
B
A
c
b
a
35
16
12
Page 607 #1-13 Every Other Odd Page 608 #17-29 Every Other Odd Page 608 #33-45 Every Other Odd
Assignments
Solve the triangle (hint: there are two solutions)
Warm-Up: May 22, 2013
C
B
A
c
b
a
28
13
8
Homework Questions?
An oblique triangle is one that does not contain a right angle
Area is one half the product of the length of two sides and the sine of the included angle
Area of an Oblique Triangle
CabArea
BacArea
AbcArea
sin2
1
sin2
1
sin2
1
Find the area of a triangle with the following measurements
Example 6: Area
22
50
20
A
feetc
feetb
Find the area of a triangle with the following measurements
You-Try #6: Area
125
5
8
B
yardsc
yardsa
Page 607 #1-13 Every Other Odd Page 608 #17-29 Every Other Odd Page 608 #33-45 Every Other Odd
Assignments