1. Make it a great day!! The choice is yours!! Complete the Do Now

2. Do Now Solve for the missing variable. 1) sin 30  = sin 60  2x 2) sin 120  = sin 45  y4 3) List the triangle congruence postulates (I know its been a long time since you took geometry!! )

3. Do Now (ans) Solve for the missing variable. 1) sin 30  = sin 60  x  3.46 2x 2) sin 120  = sin 45  y 4 x  4.90 3) List the triangle congruence postulates (I know its been a long time since you took geometry!! ) SSSSASASAAAS(not SSA!!)

4. Law of Sines Derive the law of sines Determine when to use the law of sines Derive a formula to find the area of a triangle given two sides and the included angle

5. Review Questions on Triangles How many parts are there to every triangle? How many parts do you think you MUST know in order to find the other parts? What do you know about the sum of the angles in every triangle? If you know two angles of a triangle, how can you find the third angle? How can you find the area of a triangle?

6. Law of Sines Given a  ABC, sin A = sin B = sin C a b c where a, b, and c are the lengths of the sides opposite the  A,  B and  C respectively. Note: The law of sines must use the SINE ratio!! To use the Law of Sines, one ratio (sine of angle)/(side) MUST be known. Since the Law of Sines uses three equivalent ratios, you can solve using two proportions!!

7. The Law of Sines is used for triangles which fit ASA or AAS. Why? In these two situations, you can complete one of the ratios (sine of angle)/(side)

8. Ex 1 Solve the triangle:  A = 50 ,  B = 62 , and a = 4

9. Ex 2 Solve the triangle:  A = 44 , b = 12, and a = 9

10. Area of a Triangle Given  ABC, then its area can be determined using: Area = ½ ab sin C = ½ ac sin B = ½ bc sin A Note: The angle is INCLUDED between the two sides. When will this formula be helpful to know instead of using Area = ½bh?

11. Ex 3 Find the area of the triangle:  A = 47 , b = 32 ft, c = 19 ft

12. Ex 4 Find the area of the triangle:  B = 39 , a = 15 ft, c = 21 ft

13. Warm-up (5 min.) No Talking!! Solve the triangle. 1) A = 86, a = 8, and b = 7 2) Find the area of the triangle in part A

14. How many triangles? Complete Exploration #1 on p. 480 You have 18 min. 10 min. – No Talking (Individual work) 8 min. – You may discuss with your neighbor What does the “ambiguous” case for Law of Sines mean?

15. Special Case (SSA) Given ABC, if A, a, and b (two sides and a non- included angle) are known, it is possible to form 0 triangles 1 triangle 2 triangles (one acute and one obtuse)

16. Ex 1 Determine the # of triangles formed in each situation below: a) A = 73 , a = 25, b = 28 b) C = 30 , a = 18, c = 9

17. Ex 2 Solve BOTH triangles formed if A = 64 , a = 16, and b = 17.

18. Law of Cosines Derive the law of cosines Determine when to use law of cosines Use the law of sines and the law of cosines to solve triangles and application problems

19. Warm-up (5 min.) No Talking!! Solve for the missing variable. 1) x 2 = 7 2 + 8 2 – 2(7)(8)cos 98  2) x 2 = 3 2 + 5 2 – 2(3)(5)cos 72  3) List the triangle congruence postulates for which the Law of Sines does NOT apply (I know its been a long time since you took geometry!! )

20. Warm-up (5 min.) No Talking!! Solve for the missing variable. 1) x 2 = 7 2 + 8 2 – 2(7)(8)cos 98  x  11.34 2) x 2 = 3 2 + 5 2 – 2(3)(5)cos 72  x  4.97 3) List the triangle congruence postulates for which the Law of Sines does NOT apply (I know its been a long time since you took geometry!! ) SSSSAS(not SSA either!!)

21. Law of Cosines Given  ABC, then c 2 = a 2 + b 2 – 2ab cos C, b 2 = a 2 + c 2 – 2ac cos B, a 2 = b 2 + c 2 – 2bc cos A where a, b, and c are the lengths of the sides opposite  A,  B and  C respectively. Note: The Law of Cosines uses the COSINE ratio. The Law of Cosines is used for triangles which fit the SSS or SAS situations. Why MUST you use Law of cosines? Once another angle is formed, it is easier to set up the Law of Sines ratio to find the another angle or side.

22. A closer look at the Law of Cosines Suppose  C is 90 , to what familiar formula does the Law of Cosines simplify? c 2 = a 2 + b 2 – 2ab cos 90  c 2 = a 2 + b 2 – 2ab(0) c 2 = a 2 + b 2 which is the Pythagorean theorem

23. Ex 1 Solve each triangle a) a = 3.2, b = 7.6, c = 6.4 b) B = 55 , a = 43, c = 19

24. Applying the Laws Ex 2 A parallelogram has sides of 18 and 26 ft, and an angle of 30 . Find the shorter diagonal.

25. Applying the Laws Ex 3 Find the area of a regular hexagon inscribed in a circle of radius 10 inches.

26. Applying the Laws Ex 4 Two observers are 400 ft apart on opposite sides of a tree. The angles of elevation from the observers to the top of the tree are 15° and 20°. Find the height of the tree.

27. Heron’s Formula Given a, b, and c in  ABC, the area of the triangle is given by: Area = Where s = (a+b+c)/2 Note: s is called the semiperimeter of  ABC.

28. Exit Ticket Describe how to solve triangles fitting each situation below: ASA AAS SSA SSS SAS

29. Tonight’s Assignment Check your Unit Outline Study, Study, Study!!