Angles of a Triangle. Activity: Show proof that the sum of the angles of a triangle add up to Finding the third angle of a triangle

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2 Angles of a Triangle Activity: Show proof that the sum of the angles of a triangle add up to Finding the third angle of a triangle

3 Pythagorean Theorem Is defined as the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. c 2 = a 2 + b 2 Where c = the hypotenuse

4 Using the Pythagorean Theorem to determine the length of one side of a right triangle 1. Calculate the missing side Calculate the missing side 15 5

5 What is Trigonometry? - A branch of mathematics dealing with the relationships between the sides and angles of triangles and with the relevant functions of any angles. What is a Primary Trigonometric Function? - Sine, cosine, and tangent are the primary trigonometric functions. They represent a ratio of two sides of a right triangle

6 Sides of a Triangle: There are three names for the sides of a triangle: opposite, hypotenuse, and adjacent Opposite: the side opposite (or across) to the angle you are looking at Hypotenuse: the longest side (it is always across from the right angle) Adjacent: the only one left. It is beside the angle you are looking at ** refer to the missing side worksheet and use poll everywhere

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8 Trigonometric Functions: Which one do I use? Tricks to help you remember: Soh Cah Toa OR Ships of Halifax Canadian Amateur Hockey Town of Alberton Refer to missing angle worksheet after completing the find the missing side

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10 Find the Value of the Missing Side 1. x y r 28

11 Find the Value of the Missing Angle 1. Find the value of B 2. Find the value of D Find the value of A.

12 NOTE: sin and cos can NEVER be greater than 1, but tan can be.

13 Similar Triangles Triangles are similar triangles if one triangle is an enlargement or reduction of the other The angle measure does NOT change when triangles are enlarged or reduced. How can you tell that these triangles are similar?

14 The triangles below are similar. Determine the length of the missing sides x y 8

15 Solving problems that involve more than one right triangle Example 1 Find side BD, to nearest tenth A C B D 25

16 Example 2 Find the length of AD to nearest tenth of a meter A 35.7 D B C

17 Example 3 Solve for H, to nearest tenth E A H 9.2 m B C D

18 Example 4 Solve for DC, to nearest tenth (Answer: 30.0 m) A 115 m D C B

19 Example 5 Solve for angle x, to nearest degree (Answer: 10 0 ) A 2 m D 3 m x B 10 m C

20 Example 6 Solve for x and y, to nearest tenth (Answer = 20.2 m) A x D C y 26 m B

21 Example 7 Find length of BD, to nearest tenth (Answer = 9.6m) A 20 m B D C

22 Example 8 Find length of BD, to nearest tenth A D 48.5 B 85.3 m 40.2 C

23 Example 9 Find the length of DC, to nearest tenth A m D C B

24 Interpreting word problems 1. Draw Δ ABC. If B is a right angle, C = 35 0, and the hypotenuse is 12 cm, what is the length of the side adjacent to angle C? 2. Draw Δ QRS. If Q is a right angle, find S, is the side opposite to it measures 64 cm and the side adjacent measures 56 cm.

25 Terms Acute Angle: any angle between 0 0 and 90 0 Angles of elevation or inclination are angles above the horizontal, like looking up from ground level toward the top of a flagpole. Angles of depression or declination are angles below the horizontal, like looking down from your window to the base of the building in the next lot.

26 Word Problems 1. A 10-ft ladder leans against the side of a building with its base 4 ft. from the wall. What angle, to the nearest degree, does the ladder make with the ground? (Answer: 66 0 )

27 NOTE: need better word problems with angle of elevation and angle of depression especially angle of depression like number 8 on the review sheet.

28 2. A searchlight beams shines vertically on a cloud. At a horizontal distance of 250 m from the searchlight, the angle between the ground and the line of sight to the cloud is 75 o. Determine the height of the cloud to the nearest meter. (answer 933m)

29 3. A water bomber if flying at an altitude of 5000 ft. The plane's radar shows that it is 8000 ft. from the target site on the ground. What is the angle of elevation of the plane measured from the target site, to the nearest degree? (Answer 39 0 )

30 4. A surveyor made the measurements shown in the diagram. How could the surveyor determine the distance from the transit to the survey pole to the nearest hundredth of a meter? (Answer 54.05m)

31 5. Aidan knows that the observation deck on the Vancouver Lookout is 130 m above the ground. He measures the angle between the ground and his line of sight to the observation deck is How far is Aidan from the base of the Lookout to the nearest meter? (answer: 30 m)