Lines and Angles | A

Question 12

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∘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∘39^\circ, then they score 1010 points (49∘−39∘49^\circ - 39^\circ).
  • Each team gets five turns. The winner is the team with the lowest score!
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Solution
Understand the Question

This activity is an angle estimation game played between two teams: Team 1 and Team 2.

  • One team constructs a secret angle using a protractor.
  • The other team estimates the angle without any measuring tools.
  • Points awarded equal the absolute difference between the estimated angle and the true angle.
  • After 55 rounds each, the team with the lowest total score wins.

Step 1 · Form Teams and Draw a Secret Angle

Two teams are formed: Team 1 and Team 2. Team 1 secretly chooses an angle measure (e.g., 49∘49^\circ) and draws it accurately using a protractor without Team 2 seeing it.Diagram 1

Step 2 · Estimate the Angle Measure

Team 2 views the drawn angle and discusses quickly to estimate the measure of the angle in degrees without using a protractor.Diagram 2

Step 3 · Calculate the Score

Team 1 reveals the actual angle measure using a protractor. Team 2 receives points equal to the absolute difference between their guess and the actual measure.Diagram 3

For an actual angle of 49∘49^\circ and a guess of 39∘39^\circ:

Score=∣Real angle−Guessed angle∣=49∘−39∘=10 points\begin{aligned} \text{Score} &= |\text{Real angle} - \text{Guessed angle}| \\[0.6em] &= 49^\circ - 39^\circ \\[0.6em] &= 10 \text{ points} \end{aligned}

Step 4 · Determine the Winner

Both teams take 55 turns alternately. The points from all five rounds are added up, and the team with the lowest total score wins the game.

Answer

The team with the lowest total score after 5 turns wins.

Common Mistakes
  • Score Interpretation: Thinking the team with the highest score wins; points represent error, so the lowest score wins.
  • Absolute Difference: Forgetting to take the absolute difference if the guess is greater than the true angle (e.g., guessing 55∘55^\circ for a 49∘49^\circ angle gives ∣49∘−55∘∣=6|49^\circ - 55^\circ| = 6 points, not a negative score).

More questions in A

Q1

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

Q2

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

Q3

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.

Q4

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

Q5

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 ∠AOB\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.

Q6

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

Q7

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

When folded, this is 18\dfrac{1}{8} of the circle, or 18\dfrac{1}{8} of a turn, or 18\dfrac{1}{8} of 360∘360^\circ, or 14\dfrac{1}{4} of 180∘180^\circ or 12\dfrac{1}{2} of 90∘90^\circ = ________.

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

Q8

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

Q9

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

Q10

Make the paper craft as per the given instructions. Then, unfold and open the paper fully. Draw lines on the creases made and measure the angles formed.

Q11

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.

  • ∠U=35∘\angle U = 35^\circ
  • ∠V=80∘\angle V = 80^\circ
  • ∠W=70∘\angle W = 70^\circ
  • ∠X=150∘\angle X = 150^\circ
  • ∠Y=120∘\angle Y = 120^\circ
  • ∠Z=85∘\angle Z = 85^\circ
Q12

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∘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∘39^\circ, then they score 1010 points (49∘−39∘49^\circ - 39^\circ).
  • Each team gets five turns. The winner is the team with the lowest score!
Q13

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∘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∘25^\circ, then Team 2 scores 9 points (34∘−25∘34^\circ - 25^\circ).
  • Each team gets five turns. The winner is again the team with the lowest score.
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