# Love triangles in movies

Ms. Rania Ali taught students of a primary school located in a slum.  The kids had very little interest in learning,  Ms. Ali was aware of her challenge but the great teacher found new ways to get attention of her students. This was another interesting day.

Two girls – Mehak and Arisha, were sitting in the class, chatting and giggling.  Ms, Ali asked them why they were talking in class, and what was so interesting and funny. The girls did not respond to this comment, perhaps they were scared.  Rehan, a boy sitting nearby answered instead.  He told Ms. Ali that the two of them were talking about a love triangle in a serial drama they watched last night.

Ms. Ali:  Mehak, Arisha and Rehan – all three come to the front.

Rehan:  But why me, Ms. Ali.  I was not talking.

Ms. Ali:  You were listening to them attentively enough.  So all three come to the front and face the class.  She told them to form a triangle.

# Scalene, isosceles and equilateral triangles

The whole class was giggling because they thought that this to be another love triangle.

Ms. Ali: Class, stop giggling.  The three of them are going to teach you about different types of triangles.   Do you see any relationship in the distance between Mehak and Arisha, Arisha and Rehan or Rehan and Mehak.

One of the students said that all the three distances were different from one another.

Ms. Ali:  A triangle in which all the three sides are different is called a scalene triangle. Now, I want the three of you to form a triangle in which Rehan is at the same distance from Mehak as he is from Arisha.   This type of triangle in which two sides are of equal length is called an isosceles triangle.  Notice that the angle Rehan-Mehak-Arisha is the same as Rehan-Arisha-Mehak.  This is a property of an isosceles triangle that the angles opposite to the equal sides are also equal..

The three of them were fidgeting when she told Arisha and Mehak to move so that the distance between each pair of students was the same.  This was an equilateral triangle and all three angles were also equal in it.

# Triangles ….. classified by their angles

Rajab:  Ms. Ali, My brother told me that the triangles were classified according to their angles.          Ms. Ali: Your brother was right. You can also classify triangles based on their angles.  I will draw them on the board.  If one of the angles of a triangle is 90°, it is called a right angle triangle.  In this picture, the right angle triangle is in the middle.

Rajab: The triangle on the left is kind of obese, is that why it is called obtuse?

Ms. Ali:  That’s funny Rajab.  In the triangle on the left, one angle is larger than 90° and that is why it is called an obtuse triangle.  The one on the right, has all angles less than 90° and is called an acute triangle.

Some girls in the back were whispering a cute triangle when they heard the word acute.  Suddenly, Taheen raised his hand.

Ms.  Ali: Taheen, why do you have your hand up?

Taheen:  You said that I could not draw on the school walls.  Can I draw something on a board.  It is just a bunch of lines.

Ms. Ali:  Okay Taheen but it better be good.

# Taheen just made a sketch on the board.

Ms. Ali: What did you draw Taheen?

Taheen:  I don’t know.  I just thought it would look good.  So I drew it.

Ms. Ali:  This is a picture of a triangle with its arms extended on the outside.  It makes three angles inside the triangle.  I will mark these in black as the interior angles of the triangle.  It also makes three exterior angles as I marked in red.

Ms. Ali paused for a few seconds and then said: I was going to teach you the next part another day but I will do it now because Taheen drew this picture.  Here it is. The sum of all exterior angles of a triangle is 360°.

Taheen : If I had drawn a square instead, would the sum of the exterior angles still be 360°?

Ms. Ali:  You could draw a figure with 3,4,5,6 or any number of sides, the sum of the exterior angles will always be the same: 360°.

Rajab: Ms. Ali, I see that the picture is based on three straight lines.  On each line the sum of the exterior and the interior angles will be 180°.  Then the three interior and the three exterior angles will add to 540°.  Does that mean the sum of the three interior angles of a triangle will always be 180° because 180 plus 360 is 540?

Ms. Ali:  Class, what I am doing here?  Taheen brings out a sketch to teach and then Rajab brings out the next theorem – the sum of all interior angles of a triangle is 180°.

Mehak:  I am sorry we were chatting in the back at the beginning of the class.  I want to make up by adding something – hope it is right.  A triangle cannot have more than one right angles.

Ms. Ali:  What makes you say that?

Mehak:  Because two right angles will already add up to 180° and then the third angle will have to be zero.

Arisha:  She is right.  Also an obtuse triangle can have only one angle greater than 90°.

Ms. Ali: That’s great Mehak and Arisha.  You were paying attention in the class.  I forgive you for chatting in the beginning of the class but don’t do it again.

# Rajab’s genius

Rajab came to Ms. Ali’s office afterwards and asked a question.  Would I be right if I said that the sum of all internal angles of an n sided figure will be (n x180°) – 360°?  Like for a square n =4, so the sum of all interior angles will be 360°.

Ms. Ali: You are genius Rajab.  I never thought of it that way but it is right.

# Challenge

CD is perpendicular to AB, and the lines AC and CE are equal in length.  Identify all the acute, right angle, obtuse, squalene, isosceles and equilateral triangles in this picture. Note that you may have none, one or more of any types.  No solution provided for this challenge.