Lines and Angles | FIO

Question 49

Puzzle: I am an acute angle. If you double my measure, you get an acute angle. If you triple my measure, you will get an acute angle again. If you quadruple (four times) my measure, you will get an acute angle yet again! But if you multiply my measure by 5, you will get an obtuse angle measure. What are the possibilities for my measure?

Solution

We will use the definitions of acute and obtuse angles to find the range of the unknown angle.

Step 1 — Understanding acute and obtuse angles

An acute angle is a small angle. It is always more than $\mathbf{0}$ degrees. It is always less than $\mathbf{90}$ degrees. An obtuse angle is a large angle. It is always more than $\mathbf{90}$ degrees. It is always less than $\mathbf{180}$ degrees. Let us call the mystery angle $\theta$ .

Step 2 — Using the first clues

The puzzle says the angle is acute. So, $\theta$ is less than $\mathbf{90}$ degrees. The puzzle says doubling the angle gives an acute angle. This means $2 \times \theta$ is less than $\mathbf{90}$ degrees.

$2 \times \theta < 90^\circ$

Let us divide both sides by $\mathbf{2}$ .

$\theta < \frac{90^\circ}{2}$

$\boxed{\theta < 45^\circ}$

The puzzle says tripling the angle gives an acute angle. This means $3 \times \theta$ is less than $\mathbf{90}$ degrees.

$3 \times \theta < 90^\circ$

Let us divide both sides by $\mathbf{3}$ .

$\theta < \frac{90^\circ}{3}$

$\boxed{\theta < 30^\circ}$

The puzzle says quadrupling the angle gives an acute angle. This means $4 \times \theta$ is less than $\mathbf{90}$ degrees.

$4 \times \theta < 90^\circ$

Let us divide both sides by $\mathbf{4}$ .

$\theta < \frac{90^\circ}{4}$

$\boxed{\theta < 22.5^\circ}$

From these clues, the angle $\theta$ must be less than $\mathbf{22.5}$ degrees. This is the tightest upper limit so far.

Step 3 — Using the last clue

The puzzle says multiplying the angle by $\mathbf{5}$ gives an obtuse angle. This means $5 \times \theta$ is more than $\mathbf{90}$ degrees.

$5 \times \theta > 90^\circ$

Let us divide both sides by $\mathbf{5}$ .

$\theta > \frac{90^\circ}{5}$

$\boxed{\theta > 18^\circ}$

From this clue, the angle $\theta$ must be more than $\mathbf{18}$ degrees.

Step 4 — Finding the possible range

We know the angle $\theta$ must be less than $\mathbf{22.5}$ degrees. We also know the angle $\theta$ must be more than $\mathbf{18}$ degrees. So, the angle $\theta$ is between $\mathbf{18}$ degrees and $\mathbf{22.5}$ degrees.

Answer

The possible values for the angle $\theta$ are between $\mathbf{18}^\circ$ and $\mathbf{22.5}^\circ$ . For example, the angle could be $\mathbf{19}^\circ$ , $\mathbf{20}^\circ$ , $\mathbf{21}^\circ$ , or $\mathbf{22}^\circ$ .

More questions in FIO

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d. Point P lies on $\overline{\text{AB}}$ .

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Q10

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Q11

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Q12

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Q13

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Q14

In each case, determine which angle is greater and why.

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Discuss with your friends on how you decided which one is greater.

Q15

Which angle is greater: $\angle XOY$ or $\angle AOB$ ? Give reasons.

Q16

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Q17

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Q18

Now join A to other grid points in the figure by a straight line to get a right angle. What are all the different ways of doing it?

Hint: Extend the line further as shown in the figure below. To get a right angle at A, we need to draw a line through it that divides the straight angle CAB into two equal parts.

Q19

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a. How many right angles do you have now? Justify why the angles are exact right angles.

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Q20

Identify acute, right, obtuse and straight angles in the previous figures.

Q21

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Q22

Do you know what the words acute and obtuse mean? Acute means sharp and obtuse means blunt. Why do you think these words have been chosen?

Q23

Find out the number of acute angles in each of the figures below.

What will be the next figure and how many acute angles will it have? Do you notice any pattern in the numbers?

Q24

Write the measures of the following angles:

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Making use of the medium-sized and large-sized marks, is it possible to count the number of units in 5s or 10s?

(b) $\angle\text{WAL}$

Q25

Name the different angles in the figure and write their measures.

Q26

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Q27

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Q28

Find the degree measures for the angles given below. Check if your paper protractor can be used here!

Q29

How can you find the degree measure of the angle given below using a protractor?

Q30

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Q32

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Q33

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b. What will be the angle at 2 o'clock? And at 4 o'clock? 6 o'clock?

c. Explore other angles made by the hands of a clock.

Q34

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Is it possible to express the amount by which a door is opened using an angle? What will be the vertex of the angle and what will be the arms of the angle?

Q35

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Q36

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Q37

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Q38

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Q39

Use a protractor to draw angles having the following degree measures:

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Q40

Draw an angle whose degree measure is the same as the angle given below:

Also, write down the steps you followed to draw the angle.

Q41

In each of the below grids, join A to other grid points in the figure by a straight line to get:

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b. An obtuse angle

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Mark the intended angles with curves to specify the angles. One has been done for you.

Q42

Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex.

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Q43

Draw angles with the following degree measures: a. $140^\circ$ b. $82^\circ$ c. $195^\circ$ d. $70^\circ$ e. $35^\circ$

Q44

Estimate the size of each angle and then measure it with a protractor:

a. b. c. d. e. f.

Classify these angles as acute, right, obtuse or reflex angles.

Q45

Make any figure with three acute angles, one right angle and two obtuse angles.

Q46

Draw the letter 'M' such that the angles on the sides are $40^\circ$ each and the angle in the middle is $60^\circ$ .

Q47

Draw the letter 'Y' such that the three angles formed are $150^\circ$ , $60^\circ$ and $150^\circ$ .

Q48

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Q49