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.

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

Solution

We will check each angle measurement for correct protractor use.

Step 1 — Check Angle U

The vertex U is at the center.

One arm aligns with $0^\circ$ .

The other arm points to 35°.

The inner scale is used correctly.

The angle shown is acute.

35° is an acute angle.

The student's reading is correct.

No incorrect usage is found.

$\boxed{\text{No incorrect usage}}$

Diagram 1

Step 2 — Check Angle V

The vertex V is at the center.

One arm aligns with $0^\circ$ .

The other arm points to 80°.

The inner scale is used correctly.

The angle shown is acute.

80° is an acute angle.

The student's reading is correct.

No incorrect usage is found.

$\boxed{\text{No incorrect usage}}$

Diagram 2

Step 3 — Check Angle W

The vertex W is at the center.

One arm aligns with $0^\circ$ .

The other arm points to 70°.

The outer scale is used correctly.

The angle shown is acute.

70° is an acute angle.

The student's reading is correct.

No incorrect usage is found.

$\boxed{\text{No incorrect usage}}$

Diagram 3

Step 4 — Check Angle X

The vertex is at the center.

One arm aligns with $0^\circ$ (outer scale).

The other arm points to 30° (outer scale).

The student read 150°.

This is the inner scale reading.

The student read the wrong scale.

The angle shown is acute.

150° is an obtuse angle.

The student confused the scales.

To correct, read the outer scale.

The correct angle is 30°.

$\boxed{\text{Incorrect usage: Wrong scale read}}$

Diagram 4

Step 5 — Check Angle Y

The vertex is at the center.

One arm aligns with $0^\circ$ (inner scale).

The other arm points to 60° (inner scale).

The student read 120°.

This is the outer scale reading.

The student read the wrong scale.

The angle shown is acute.

120° is an obtuse angle.

The student confused the scales.

To correct, read the inner scale.

The correct angle is 60°.

$\boxed{\text{Incorrect usage: Wrong scale read}}$

Diagram 5

Step 6 — Check Angle Z

The vertex is at the center.

One arm is not on the $0^\circ$ line.

It is on the $90^\circ$ mark.

The protractor is placed incorrectly.

One arm must align with $0^\circ$ .

The other arm points to $145^\circ$ (inner scale).

The angle shown is $145^\circ - 90^\circ = \textbf{55°}$ .

The student read 85°.

This is a misreading or miscalculation.

To correct, align one arm with $0^\circ$ .

Then read the correct scale.

$\boxed{\text{Incorrect usage: Protractor misaligned}}$

Diagram 6

Answer

(i) $\angle U = 35^\circ$ : No incorrect usage. The protractor is used correctly. (ii) $\angle V = 80^\circ$ : No incorrect usage. The protractor is used correctly. (iii) $\angle W = 70^\circ$ : No incorrect usage. The protractor is used correctly. (iv) $\angle X = 150^\circ$ : Incorrect usage. The student read the wrong scale. The angle shown is 30°. To correct, read the outer scale. (v) $\angle Y = 120^\circ$ : Incorrect usage. The student read the wrong scale. The angle shown is 60°. To correct, read the inner scale. (vi) $\angle Z = 85^\circ$ : Incorrect usage. The protractor is misaligned. One arm is not on the $0^\circ$ line. The angle shown is 55°. To correct, align one arm with $0^\circ$ .

More questions in A

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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.

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

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

When folded, this is $\frac{1}{8}$ of the circle, or $\frac{1}{8}$ of a turn, or $\frac{1}{8}$ of $360^\circ$ , or $\frac{1}{4}$ of $180^\circ$ or $\frac{1}{2}$ 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 OB, OC, ..., 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!

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

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° 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°, then they score 10 points (49°–39°).
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°.
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°, then Team 2 scores 9 points (34°–25°).
Each team gets five turns. The winner is again the team with the lowest score.