Lines and Angles | A

Question 11

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
Question diagram 1
Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question

To use a protractor correctly to measure an angle:

  • Place the centre point (origin) of the protractor directly on the vertex of the angle.
  • Align the baseline (0∘0^\circ line) with one arm of the angle.
  • Use the scale (inner or outer) where the baseline starts at 0∘0^\circ.
  • Read the measurement where the second arm crosses that same scale.
  • Visually verify whether the angle is acute (<90∘< 90^\circ) or obtuse (>90∘> 90^\circ) as a sanity check.

(i) ∠U=35∘\angle U = 35^\circ

Step 1 · Check Alignment and Scale for ∠U\angle U

Diagram 1

  • Vertex: Centred at UU.
  • Baseline: One arm aligns with 0∘0^\circ on the inner scale.
  • Reading: The second arm points directly to 35∘35^\circ on the inner scale.
  • Observation: The angle is acute, matching the reading of 35∘35^\circ.
Answer

(i) No incorrect usage. The protractor is used correctly; ∠U=35∘\angle U = 35^\circ.

(ii) ∠V=80∘\angle V = 80^\circ

Step 1 · Check Alignment and Scale for ∠V\angle V

Diagram 2

  • Vertex: Centred at VV.
  • Baseline: One arm aligns with 0∘0^\circ on the inner scale.
  • Reading: The second arm points to 80∘80^\circ on the inner scale.
  • Observation: The angle is acute, matching the reading of 80∘80^\circ.
Answer

(ii) No incorrect usage. The protractor is used correctly; ∠V=80∘\angle V = 80^\circ.

(iii) ∠W=70∘\angle W = 70^\circ

Step 1 · Check Alignment and Scale for ∠W\angle W

Diagram 3

  • Vertex: Centred at WW.
  • Baseline: One arm aligns with 0∘0^\circ on the outer scale (left side).
  • Reading: The second arm points to 70∘70^\circ on the outer scale.
  • Observation: The angle is acute, matching the reading of 70∘70^\circ.
Answer

(iii) No incorrect usage. The protractor is used correctly; ∠W=70∘\angle W = 70^\circ.

(iv) ∠X=150∘\angle X = 150^\circ

Step 1 · Identify Mistake and Correction for ∠X\angle X

Diagram 4

  • Incorrect usage: The student read 150∘150^\circ from the inner scale instead of the outer scale.
  • Correction: Since the baseline arm aligns with 0∘0^\circ on the outer scale, follow the outer scale. Correct measure of ∠X=30∘\text{Correct measure of } \angle X = 30^\circ
Answer

(iv) Incorrect usage: Wrong scale read. The correct angle is 30∘30^\circ.

(v) ∠Y=120∘\angle Y = 120^\circ

Step 1 · Identify Mistake and Correction for ∠Y\angle Y

Diagram 5

  • Incorrect usage: The student read 120∘120^\circ from the outer scale instead of the inner scale.
  • Correction: Since the baseline arm aligns with 0∘0^\circ on the inner scale, follow the inner scale. Correct measure of ∠Y=60∘\text{Correct measure of } \angle Y = 60^\circ
Answer

(v) Incorrect usage: Wrong scale read. The correct angle is 60∘60^\circ.

(vi) ∠Z=85∘\angle Z = 85^\circ

Step 1 · Identify Mistake and Correction for ∠Z\angle Z

Diagram 6

  • Incorrect usage: The protractor is misaligned; the base arm is placed along the 90∘90^\circ line instead of the 0∘0^\circ line.
  • Angle calculation from figure: Angle=145∘−90∘=55∘\text{Angle} = 145^\circ - 90^\circ = 55^\circ
  • Correction: Align one arm of the angle with the 0∘0^\circ baseline and read the resulting position on that scale.
Answer

(vi) Incorrect usage: Protractor misaligned. The correct angle is 55∘55^\circ.

Common Mistakes
  • Reading the Wrong Scale: Reading the inner scale instead of the outer scale (or vice versa) gives the supplementary angle (180∘−θ180^\circ - \theta). Always start from the scale that has 0∘0^\circ on the initial arm.
  • Misaligning the Baseline: Placing the edge or the 90∘90^\circ mark on an arm instead of aligning the 0∘0^\circ baseline with one arm.
  • Not Checking Angle Type: Forgetting to check whether the angle is visibly acute (<90∘< 90^\circ) or obtuse (>90∘> 90^\circ) to catch scale errors.

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.
← Back to Lines and Angles