An openning story to the stars

### Junior Geometry #### An openning story to the stars Our story start from the straight lines, You might look around the place you are sitting, Can you see how we, human being, are so addictive to the straight line? There’s no escaping the straight line on our planet. --- ### Chapter 1: #### Two straight lines on a plane There is three relationships between two straight lines in on a plane: - Parallel to each other - Intersecting with angles - Coincident lines: they lies on top of each other <img src="https://i.ibb.co/yNrB97T/Sharing1.png" width="600" height="200"> --- ### Chapter 1: #### Then Three ... "Z", "F", "C" When one straight line cross two parallel lines (the line is called '**transversal**') we have the **'ZFC' theorem**. <img src="https://i.ibb.co/k4r6MGS/Sharing2.png" width="680" height="300"> --- ### Chapter 1: #### $180^o$ or $\pi$ When three straight lines form a triangle, the sum of interior angles is $180^o$ - Can you prove this classic **"fact"** using the **ZFC Theorem**? <img src="https://i.ibb.co/sjFVvTZ/Sharing3.png" width="500" height="300"> --- ### Chapter 2: 5 Boards! We are good at this: Create idea or object from scratch, you might call it "creativity". - It is time to create some interesting "facts" based on our knowledge so far. - I will show you the first one, then you might "borrow" my method to do the rest <img src="https://i.ibb.co/qYmq2zL/Sharing4.png" width="500" height="300"> - Board No.1: if $AB \parallel CD$, then $\alpha = \beta + \gamma$. No matter how I might move $E$. - Show it: construct a line through E and $\parallel AB$, can you see **Z-theorem** now? --- ### Chapter 2: 5 Boards! Now your turn to the 2nd board <img src="https://i.ibb.co/qj7w2Jr/Sharing5.png" width="500" height="300"> - Board No.2: if $AB \parallel CD$, then $\alpha + \beta + \gamma = 360^o$. No matter how I might move $E$. - Show it: Your turn? "Z" or "F" or "C" ? --- ### Chapter 2: 5 Boards! The 3rd board <img src="https://i.ibb.co/j5PNZ3S/Sharing6.png" width="500" height="300"> - Board No.3: if $AB \parallel CD$, then $\beta = \alpha + \gamma$. No matter how I might move $E$. - Show it: Remember to "copy" my method? you might try different constructions? --- ### Chapter 2: 5 Boards! The 4th board <img src="https://i.ibb.co/tzVYJ7p/Sharing7.png" width="500" height="200"> - Board No.4: if $AB \parallel CD$, then the sum of red angles must equals to the sum of blue angles. - Show it: You might "copy" my method several times. --- ### Chapter 2: 5 Boards! The last board <img src="https://i.ibb.co/XYqz0g6/Sharing8.png" width="500" height="200"> - Board No.5: if $AB \parallel CD$, then the sum of interior angles $=180^o \times n$, where n is number of segments between two parallel lines. In the graph above, we have 4 segments, sum of 5 interior angles must equals to $180 \times 4 = 720^o$ - Show it: You might "copy" my method several times. --- ### Interlude: To the star We saw the beauty and power of creativity, sometimes, draw a line from nothing will help us a lot in solving problems Do you still remember the 1st board? <img src="https://i.ibb.co/vVLpD45/Sharing9.png" width="500" height="200"> - A Dart: from the board no.1, we let two parallel lines close down and meet, then we will have a new shape. We might call it **"Dart"** for now. - Another beautiful feature of this shape is that, the exterior angle $\alpha$, equals to sum of three interior angles $\beta$, $\gamma$, and D - Can you show this? --- ### Overture: The stars Now comes our first main charactor <img src="https://i.ibb.co/HxB4X0Y/Sharing10.png" width="300" height="200"> - A Star: More precisely, a star-polygon. - A beautiful feature: Angles at 5 vertices, add up to $180^o$. - Can you show this? Hint: you might need the beautiful feature from the dart. --- ### Overture: The stars Things getting more interesting .. The previous one, Mathematically, we call it $(5,2)$ star-polygon <img src="https://i.ibb.co/BK0kPCK/Sharing11.png" width="300" height="200"> - This one, we call it (7,3) Star Polygon - Now this one has 7 vertical angles, guess what is the sum of these 7 angles? - $180^o$ again! Can you prove it? --- ### Overture: Draw the stars We are not going too far today.. But before we finish, can you draw this star on a paper from scratch? <img src="https://i.ibb.co/BK0kPCK/Sharing11.png" width="300" height="200"> - Hmm, if you failed, think about its name, (7,3) Star Polygon, what is 7 stand for, and what does 3 means? - Well, if you succeed, you might try other combination of numbers, for example, (6, 3), is there a (6, 3) star-polygon? --- ### Finale: Co-Prime I believe you have tried some combination of numbers. Yon might find that: Only if two numbers, (n, m) are **co-prime**, there is a star-polygon! - What is co-prime? Well for two positive integers, $n, m$, we say that $n$ and $m$ are co-prime if and only if $gcd (n, m) = 1$, i.e. the greatest common divisor of two numbers is 1. - Of course, they don't need to be both prime, for example, $(9, 4)$ is a pair of co-prime. --- ### Finale: The name We are going to end our story here, keep it openning for you to explore. We started from straight lines, and went to the **Stars** and **Numbers**, What a journey! If you have learnt something from this story, it may deserve a name. How about, **"Twinkle Twinkle Little Star"** :slightly_smiling_face: By SU, 15/06/2023 @TGS