**Sun always rises in the east**

Ms Rania Ali entered her class, She was in a great mood and asked the class: In which direction did the sun rise today ?

The class was confused. Was there was something unique about today ? Yet, the students did not have the courage to ask.

Finally Taheen stood up and said: Ms. Ali, the sun rose in the East as it does every other day. Did I miss something ? Is something special about today ?

Ms. Ali: Sorry class. It seems that I confused you by using the word today in my question. As Taheen says that the sun always rises in the East. If you agree please raise your hands.

Everyone in the class raised their hands.

Ms. Ali: Farah, please come to the front and stand with both hands stretched.

Farah stood in front of the class but Ms. Ali said: I am going to ask Farah to turn so that her right arm is towards East. In which direction does her left arm point ? (see the picture)

The class said West.

Ms. Ali: That was easy. What about her nose ?

A few students said that Farah’s nose pointed towards North, and they added that her back was towards South.

**Directions and angles**

Ms. Ali: Everybody remembers that the rotation of a full circle makes an angle of 360⁰. Rehan, then what is the angle between East and North ?

Rehan: The angle between East and North is one fourth of a full circle, and this angle will, therefore. be 360⁰/4 or 90⁰.

Mehak: Does that mean the angles between North and West, West and South and South and East will also be 90⁰ ?

Ms. Ali: Yes Mehak. What is the angle between East and West ?

Mehak: That would be half of the full circle or 180⁰.

Ms. Ali: I have something interesting to show you. It is called a compass. You can put it anywhere and it shows you the different directions.

Taheen: I see, it not only shows the four directions but also divides them further, like you have E, ENE, EN, NNE and N (see the picture below).

Ms. Ali: Taheen, what do you think the angle between E and EN will be ?

Taheen: The angle between E and N is 90⁰. So the angle between E and NE will be half that which is 45⁰, and the angle between E and ENE or between ENE and NE will be half of that which is 22.5⁰.

Mehak: So the compass divides the circle of directions into 16 angles each of which is 1/16th part of 360⁰ which comes out to 22.5⁰.

Faizal: My uncle has a compass in this car. It looks a little different. It is in the front window of the car, and it shows directions both in letters like N, NW, W and in angles from North in degrees from 0 to 360⁰. When the car turns left or right, the angle changes by 90⁰. It is neat because it helps him check which direction the car is going in.

**Measuring an angle**

Fatima: Ms. Ali, you showed us how to calculate an angle based on what fraction of a circle. How do you measure any angle without knowing how much part of a circle it makes ?

Ms. Ali: Your class is very lucky. A lady donated these geometry boxes for you. I will pass these on. Each student will get one. You can take it home but if you lose it, you will have to buy one on your own.

Ms. Ali gave one box to each student and then said: In this box you have a D-shaped structure which is called a protractor. The D has a line at the bottom, a tiny circle at the centre and the angles written on the semicircle of the D. These angles go from 0 to 180⁰.

Let us say, you want to measure the angle BAC (see the picture below). Place the bottom line of the D on the line AB. In this picture the line AC crosses the D at 25⁰, So the angle BAC is 25⁰. You can measure any angle between 0 and 180⁰ with this instrument.

As always, Taheen found a new game and seems amused. He was playing with the newly found D.

Ms. Ali: Taheen, why are you smiling ? What is so amusing ? Tell us all.

Taheen: I found a new game with this D. You told us to place the line AB at 0⁰ and the line AC came to be at angle of 25⁰. I can rotate the D so that the line AB lies on it at a different angle. When I put AB at the angle 30⁰, the line AC crosses the D at 55⁰. The difference between 55 and 30 is 25⁰, the same number for the angle you showed us by your method. It is fun to play with the D.

**Challenge**

Use the D to draw angles of 10, 35, 60, 80, 120 and 160⁰