Find the missing parts of each triangle. Students, Take out your calendar and your homework. Take out your spiral notebook and Complete the DNA. Use your notes if necessary. Find the missing parts of each triangle. 1) Find the measure of each angle. 2)

Digital Lesson Law of Sines

Definition: Oblique Triangles An oblique triangle is a triangle that has no right angles. C B A a b c To solve an oblique triangle, you need to know the measure of at least one side and the measures of any other two parts of the triangle – two sides, two angles, or one angle and one side. Definition: Oblique Triangles

Solving Oblique Triangles The following cases are considered when solving oblique triangles. Two angles and any side (AAS or ASA) A C c A B c 2. Two sides and an angle opposite one of them (SSA) C c a 3. Three sides (SSS) a c b c a B 4. Two sides and their included angle (SAS) Solving Oblique Triangles

Definition: Law of Sines The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.) Law of Sines If ABC is an oblique triangle with sides a, b, and c, then C B A b h c a C B A b h c a Acute Triangle Obtuse Triangle Definition: Law of Sines

Example: Law of Sines - ASA Example (ASA): Find the remaining angle and sides of the triangle. C B A b c 60 10 a = 4.5 ft The third angle in the triangle is A = 180 – A – B = 180 – 10 – 60 = 110. 4.15 ft 110 0.83 ft Use the Law of Sines to find side b and c. Example: Law of Sines - ASA

Example: Single Solution Case - SSA Example (SSA): Use the Law of Sines to solve the triangle. A = 110, a = 125 inches, b = 100 inches C B A b = 100 in c a = 125 in 110 21.26 48.74 48.23 in C  180 – 110 – 48.74 = 21.26 Example: Single Solution Case - SSA

Example: No-Solution Case - SSA Example (SSA): Use the Law of Sines to solve the triangle. A = 76, a = 18 inches, b = 20 inches C A B b = 20 in a = 18 in 76 There is no angle whose sine is 1.078. There is no triangle satisfying the given conditions. Example: No-Solution Case - SSA

Area of an Oblique Triangle C B A b c a Find the area of the triangle. A = 74, b = 103 inches, c = 58 inches Example: 103 in 74 58 in Area of an Oblique Triangle

The flagpole is approximately 9.5 meters tall. Application: A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14 with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20. 20 A 70 Flagpole height: b 34 B 16 m C 14 The flagpole is approximately 9.5 meters tall. Application

Complete each identity.

Example: Two-Solution Case - SSA Example (SSA): a = 11.4 cm C A B1 b = 12.8 cm c 58 Use the Law of Sines to solve the triangle. A = 58, a = 11.4 cm, b = 12.8 cm 49.8 72.2 10.3 cm C  180 – 58 – 72.2 = 49.8 Two different triangles can be formed. Example continues. Example: Two-Solution Case - SSA

Example: Two-Solution Case – SSA continued Example (SSA) continued: 72.2 10.3 cm 49.8 a = 11.4 cm C A B1 b = 12.8 cm c 58 Use the Law of Sines to solve the second triangle. A = 58, a = 11.4 cm, b = 12.8 cm B2  180 – 72.2 = 107.8  C  180 – 58 – 107.8 = 14.2 C A B2 b = 12.8 cm c a = 11.4 cm 58 14.2 107.8 3.3 cm Example: Two-Solution Case – SSA continued