# Ms. Ali caught Taheen writing graffiti

Ms. Rania Ali taught in a primary school located in a slum.  The kids had very little interest in learning.  Ms. Rania Ali was aware of her challenge but the great teacher found new ways to get attention of her students. This is the story of the time when she caught Taheen – a creative but mischievous boy writing some graffiti on the school wall.

After school, most kids went home except for some who just hung around or went to different places to roam.  Taheen took a chalk from the school and was writing some graffiti on the school wall.  Nothing serious – he was just drawing lines on the wall.  The teacher Ms. Rani Ali saw him and Taheen ran away.

The next day in class Ms. Ali called out Taheen’s name.  Now, the boy was scared because he knew what he had done the day before and also that the teacher had seen him. Nevertheless Taheen responded, “Yes, Ms. Ali”.

# Draw lines exactly the same way on the blackboard

Ms. Ali: Taheen, I saw you drawing lines on the school wall.  Here is a chalk.  Draw lines exactly the same way on the blackboard.

Taheen drew only two lines when Ms. Ali stopped him.

Ms.  Ali:  Why did you draw the two lines this way?  What is unique about them?

Taheen (shy and unsure): The distance between them is the same throughout.

Some of the kids were whispering that he had drawn parallel lines.

Ms. Ali: Why are you all whispering?  Just say he drew two parallel lines because that is right. Where have you seen parallel lines before?

Mehak:  Railway tracks are parallel lines because the same train has to go on them all the way.  Edges of some roads are also parallel.

Rajab:  My uncle bought some wood, the edges of the wood were parallel. Also I have seen houses with brick walls.  Space between the bricks form parallel lines.

Ms. Ali:  That’s good, wherever, the distance between two lines remains the same, the lines are parallel.

Then she took the chalk and drew a line across the two parallel lines.  At first, the students were confused why Ms. Ali would cross out the parallel lines this way but then she spoke.

# When a line crosses two parallel lines

Ms. Ali:  Thanks to Taheen’s lines, you will learn about different angles when another line crosses two parallel lines.  I have written names of the angles.  How many of you can figure out angle B, angle C and angle D if you were given angle A?
Many students raised hands but she called on Fateh.

Fateh: We know that sum of all angles on a straight line is 180°.  Therefore, angle B = 180° – angle A  and also angle D = 180° – angle A.  Because, it is the same straight lines form the opposite angles A and C, they have to be the same.

Ms. Ali: Very good Fateh.  We can also say that angle B and angle D are supplementary to angle A.  Angles A and C are vertical or opposite angles.  Now Taheen has to tell us the relationship between angles E and A.

Taheen: We have not yet learned this.  So I am going to guess.  Are they equal?

Ms. Ali:  You are right Taheen. Angles E and A are equal.  They are also called the corresponding angles.  Now, Arisha how are angles A and G related?

Arisha:  They should be equal. Angles G and E are equal being vertical angles and angles E and A are equal being the corresponding angles.  So angles G and A would also be equal.

Ms. Ali:  Yes, and these are called alternate interior angles.

Rehan was sitting in the back playing with his pencil as if he was in a band when Ms. Ali called his name.  He was really surprised.

Rehan: Yes, Ms. Ali. You just told us about alternate interior angles.  I was thinking then we must also have alternate exterior angles.

Ms. Ali: I see you were paying attention.  Very good, the alternate exterior angles would be like angles F and D.  These would also be equal.

Alia: I think Taheen is very smart.  He drew this graffiti just to teach us about these angles.

Taheen:  Thank you Alia.   Now, I have to think of what to do for next time.

Ms. Ali:  Taheen, no more graffiti or else I will send you to the principal’s office.

# Challenge

Taheen had drawn one line on the wall and another line above it and parallel to it.  Then on the top, he drew a third line which was parallel to the second one.  Someone drew a line that crossed these lines and wants to know if the blue angle is the same as the red angle.

No solution is provided but here is the hint: The third line is parallel to the first line because a line to a parallel line is also parallel.