1. 7.1 The Law of Sines Congruence Axioms
Side-Angle-Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.
Angle-Side-Angle (ASA) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.
Side-Side-Side (SSS) If three sides of one triangle are equal to three sides of a second triangle, the triangles are congruent.

2. 7.1 Data Required for Solving Oblique Triangles
Case 1 One side and two angles known:
SAA or ASA
Case 2 Two sides and one angle not included
between the sides known:
SSA
This case may lead to more than one solution.
Case 3 Two sides and one angle included between
the sides known:
SAS
Case 4 Three sides are known:
SSS

3. 7.1 Derivation of the Law of Sines
Start with an acute or obtuse triangle
and construct the perpendicular from
B to side AC. Let h be the height of this
perpendicular. Then c and a are the
hypotenuses of right triangle ADB
and BDC, respectively.

4. 7.1 The Law of Sines
In a similar way, by constructing perpendiculars from
other vertices, the following theorem can be proven.
Alternative forms are sometimes convenient to use:
Law of Sines
In any triangle ABC, with sides a, b, and c,

5. 7.1 Using the Law of Sines to Solve a Triangle
Example Solve triangle ABC if A = 32.0°, B = 81.8°,
and a = 42.9 centimeters.
Solution Draw the triangle
and label the known values.
Because A, B, and a are known,
we can apply the law of sines involving these variables.

6. 7.1 Using the Law of Sines to Solve a Triangle
To find C, use the fact that there are 180° in a triangle.
Now we can find c.

7. 7.1 Using the Law of Sines in an Application (ASA)
Example Two stations are on an east-west
line 110 miles apart. A forest fire is located
on a bearing of N 42° E from the western
station at A and a bearing of N 15° E from
the eastern station at B. How far is the fire
from the western station?
Solution Angle BAC = 90° – 42° = 48°
Angle B = 90° + 15° = 105°
Angle C = 180° – 105° – 48° = 27°
Using the law of sines to find b gives

8. 7.1 Ambiguous Case
If given the lengths of two sides and the angle opposite one of them, it is possible that 0, 1, or 2 such triangles exist.
Some basic facts that should be kept in mind:
For any angle , –1  sin   1, if sin  = 1, then
 = 90° and the triangle is a right triangle.
sin  = sin(180° –  ).
The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-value angle is opposite the intermediate side (assuming unequal sides).

9. 7.1 Ambiguous Case

10. 7.1 Ambiguous Case for Obtuse Angle A

11. 7.1 Solving the Ambiguous Case: No Such Triangle
Example Solve the triangle ABC if B = 55°40´,
b = 8.94 meters, and a = 25.1 meters.
Solution Use the law of sines to find A.
Since sin A cannot be greater than 1, the triangle does
not exist.

12. 7.1 Solving the Ambiguous Case: Two Triangles
Example
Solve the triangle ABC if A = 55.3°,
a = 22.8 feet, and b = 24.9 feet.
Solution

13. 7.1 Solving the Ambiguous Case: Two Triangles
To see if B2 = 116.1° is a valid possibility, add 116.1°
to the measure of A: 116.1° ° = 171.4°. Since
this sum is less than 180°, it is a valid triangle.
Now separate the triangles into two: AB1C1 and AB2C2.

14. 7.1 Solving the Ambiguous Case: Two Triangles
Now solve for triangle AB2C2.

15. 7.1 Number of Triangles Satisfying the Ambiguous Case
Let sides a and b and angle A be given in triangle ABC. (The
law of sines can be used to calculate sin B.)
If sin B > 1, then no triangle satisfies the given conditions.
If sin B = 1, then one triangle satisfies the given conditions and B = 90°.
If 0 < sin B < 1, then either one or two triangles satisfy the given conditions
If sin B = k, then let B1 = sin-1 k and use B1 for B in the first triangle.
Let B2 = 180° – B1. If A + B2 < 180°, then a second triangle exists. In this case, use B2 for B in the second triangle.

16. 7.1 Solving the Ambiguous Case: One Triangle
Example Solve the triangle ABC, given A = 43.5°,
a = 10.7 inches, and c = 7.2 inches.
Solution
The other possible value for C:
C = 180° – 27.6° = 152.4°.
Add this to A: 152.4° ° = 195.9° > 180°
Therefore, there can be only one triangle.

17. 7.1 Solving the Ambiguous Case: One Triangle