Question 10

Fold a piece of paper and unfold it. Do you see a crease?

Mark any two points $A$ and $B$ on a sheet of paper. Try to connect $A$ to $B$ by various routes (Fig. 2.1). What is the shortest route from $A$ to $B$?

2.7 Making Rotating Arms

Make several 'rotating arms' with different angles between the arms. Arrange the angles you have made from smallest to largest by comparing and using superimposition.

Now, shuffle and mix up all the rotating arms. Can you identify which of the rotating arms will pass through the slit?

Let's Explore

We can try to solve this problem using a piece of paper. Recall that when a fold is made, it creates a crease which is straight.

Take a rectangular piece of paper and on one of its sides, mark the straight angle AOB. By folding, try to get a line (crease) passing through O that divides $\angle AOB$ into two equal angles.

How can it be done?

Fold the paper such that OB overlaps with OA. Observe the crease and the two angles formed.

Fold the semi-circular sheet in half as shown in Fig. 2.15 to form a quarter circle.

Fold the sheet again as shown in Figs. 2.16 and 2.17:

When folded, this is $\dfrac{1}{8}$ of the circle, or $\dfrac{1}{8}$ of a turn, or $\dfrac{1}{8}$ of $360^\circ$, or $\dfrac{1}{4}$ of $180^\circ$ or $\dfrac{1}{2}$ of $90^\circ$ = ________.

The new creases formed give us measures of $45^\circ$ and $180^\circ - 45^\circ = 135^\circ$ as shown. Write $45^\circ$ and $135^\circ$ at the correct places on the new creases along the edge of the semicircle.

Continuing with another half fold as shown in Fig. 2.18, we get an angle of measure ________

Unfold and mark the creases as $\text{OB}$, $\text{OC}$, $\dots$, etc., as shown in Fig. 2.19 and Fig. 2.20.

Mind the Mistake, Mend the Mistake!

A student used a protractor to measure the angles as shown below. In each figure, identify the incorrect usage(s) of the protractor and discuss how the reading could have been made and think how it can be corrected.

• $\angle U = 35^\circ$
• $\angle V = 80^\circ$
• $\angle W = 70^\circ$
• $\angle X = 150^\circ$
• $\angle Y = 120^\circ$
• $\angle Z = 85^\circ$

Let's Play a Game #1

This is an angle guessing game! Play this game with your classmates by making two teams, Team 1 and Team 2. Here are the instructions and rules for the game:

• Team 1 secretly choose an angle measure, for example, $49^\circ$ and makes an angle with that measure using a protractor without Team 2 being able to see it.
• Team 2 now gets to look at the angle. They have to quickly discuss and guess the number of degrees in the angle (without using a protractor!).
• Team 1 now demonstrates the true measure of the angle with a protractor.
• Team 2 scores the number of points that is the absolute difference in degrees between their guess and the correct measure. For example, if Team 2 guesses $39^\circ$, then they score $10$ points ($49^\circ - 39^\circ$).
• Each team gets five turns. The winner is the team with the lowest score!

Let's Play a Game #2

We now change the rules of the game a bit. Play this game with your classmates by again making two teams, Team 1 and Team 2. Here are the instructions and rules:

• Team 1 announces to all, an angle measure, e.g., $34^\circ$.
• A player from Team 2 must draw that angle on the board without using a protractor. Other members of Team 2 can help the player by speaking words like ‘Make it bigger!’ or ‘Make it smaller!’.
• A player from Team 1 measures the angle with a protractor for all to see.
• Team 2 scores the number of points that is the absolute difference in degrees between Team 2’s angle size and the intended angle size. For example, if player’s angle from Team 2 is measured to be $25^\circ$, then Team 2 scores 9 points ($34^\circ - 25^\circ$).
• Each team gets five turns. The winner is again the team with the lowest score.