Is it possible to draw such that the two angles are equal to each other in size?

Understand the Question A straight line forms a straight angle measuring . A ray drawn from a point on line divides this straight angle into two adjacent angles: and . To make the two angles equal, their sum must still be , which means each angle must measure (making perpendicular to line ). Step 1 · Calculate the Measure of the Equal Angles A straight angle on line measures . Let the two equal angles be . Step 2 · Determine the Position of Ray Since each angle is (a right angle), ray must be drawn perpendicular to line (). Therefore, it is possible to draw such a ray. Answer > Yes, it is possible. When ray is perpendicular to line , both angles are equal to . Common Mistakes Assuming Any Angle Works: For the two angles on a straight line to be equal, both must be exactly ; they cannot be acute or obtuse. Incorrect Angle Sum: Forgetting that angles on a straight line at a point always sum to , not or .

Lines and Angles | IT

Question 12

Is it possible to draw $\overrightarrow{OC}$ such that the two angles are equal to each other in size?

Solution

Understand the Question

A straight line forms a straight angle measuring $180^\circ$ .
A ray $\overrightarrow{OC}$ drawn from a point $O$ on line $AB$ divides this straight angle into two adjacent angles: $\angle AOC$ and $\angle BOC$ .
To make the two angles equal, their sum must still be $180^\circ$ , which means each angle must measure $90^\circ$ (making $\overrightarrow{OC}$ perpendicular to line $AB$ ).

Step 1 · Calculate the Measure of the Equal Angles

A straight angle on line $AB$ measures $180^\circ$ . Diagram 1

$\angle AOC + \angle BOC = 180^\circ$

Let the two equal angles be $\angle AOC = \angle BOC = x$ .

\begin{aligned} x + x &= 180^\circ \\[0.6em] 2x &= 180^\circ \\[0.6em] x &= \dfrac{180^\circ}{2} \\[0.6em] x &= 90^\circ \end{aligned}

Step 2 · Determine the Position of Ray $\overrightarrow{OC}$

Since each angle is $90^\circ$ (a right angle), ray $\overrightarrow{OC}$ must be drawn perpendicular to line $AB$ ( $\overrightarrow{OC} \perp AB$ ).

Therefore, it is possible to draw such a ray.

Answer

Yes, it is possible. When ray $\overrightarrow{OC}$ is perpendicular to line $AB$ , both angles are equal to $90^\circ$ .

Common Mistakes

Assuming Any Angle Works: For the two angles on a straight line to be equal, both must be exactly $90^\circ$ ; they cannot be acute or obtuse.
Incorrect Angle Sum: Forgetting that angles on a straight line at a point always sum to $180^\circ$ , not $360^\circ$ or $90^\circ$ .

More questions in IT

Do you think you can draw a complete picture of a line? No. Why?

Do you see angles being made in each of these cases? Can you mark their arms and vertex?

Which angle is greater—the angle in Case 1 or the angle in Case 2?

In a compass or divider, we turn the arms to form an angle. The vertex is the point where the two arms are joined. Identify the arms and vertex of the angle.

A pair of scissors has two blades. When we open them (or 'turn them') to cut something, the blades form an angle. Identify the arms and vertex of the angle.

Look at the pictures of spectacles, wallet and other common objects. Identify the angles in them by marking out their arms and vertices.

Look at these animals opening their mouths. Do you see any angles here? If yes, mark the arms and vertex of each one. Some mouths are open wider than others; the more the turning of the jaws, the larger the angle! Can you arrange the angles in this picture from smallest to largest?

Is it always easy to compare two angles?

Here are some angles. Label each of the angles. How will you compare them? Draw a few more angles; label them and compare.

Suppose we have a transparent circle which can be moved and placed on figures. Let us place the circular paper on the angle made by the first crane. The circle is placed in such a way that its centre is on the vertex of the angle. Let us mark the points $A$ and $B$ on the edge circle at the points where the arms of the angle pass through the circle.

Q. Can we use this to find out if this angle is greater than, or equal to or smaller than the angle made by the second crane?

Q10

Let us place the circle on the angle made by the second crane so that the vertex coincides with the centre of the circle and one of the arms passes through $OA$ .

Q. Can you now tell which angle is bigger?

Q11

Which crane was making the bigger angle?

If you can make a circular piece of transparent paper, try this method to compare the angles in Fig. 2.10 with each other.

Q12

Is it possible to draw $\overrightarrow{OC}$ such that the two angles are equal to each other in size?

Q13

If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?

Q14

Angles are classified in three groups as shown below. Right angles are shown in the second group. What could be the common feature of the other two groups?

Q15

What is the measure of a straight angle in degrees? A straight angle is half of a full turn. As a full-turn is $360^\circ$ , a half turn is $180^\circ$ . What is the measure of a right angle in degrees? Two right angles together form a straight angle. As a straight angle measures $180^\circ$ , a right angle measures $90^\circ$ .

Q16

The circle has been divided into 1, 2, 3, 4, 5, 6, 8, 9 10 and 12 parts below. What are the degree measures of the resulting angles? Write the degree measures down near the indicated angles.

Q17

There are two sets of numbers on the protractor: one increasing from right to left and the other increasing from left to right. Why does it include two sets of numbers?

Q18

What is the degree measure of $\angle \text{AOB}$ ?

Q19

In Fig. 2.19, we have $\angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOG = \angle GOH = \angle HOI = \text{______}$ . Why?

Q20

Measure all three angles of the triangle shown in Fig. 2.21 (a), and write the measures down near the respective angles. Now add up the three measures. What do you get? Do the same for the triangles in Fig. 2.21 (b) and (c). Try it for other triangles as well, and then make a conjecture for what happens in general! We will come back to why this happens in a later year.

Q21

Let's Explore

In this figure, $\angle \text{TER} = 80^\circ$ . What is the measure of $\angle \text{BET}$ ? What is the measure of $\angle \text{SET}$ ?

Hint: Observe that $\angle \text{REB}$ is a straight angle. Hence, the degree measure of $\angle \text{REB} = 180^\circ$ of which $80^\circ$ is covered by $\angle \text{TER}$ . A similar argument can be applied to find the measure of $\angle \text{SET}$ .