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Straight Angles

Introduction

We know that an angle is a portion of the plane between two rays that are joined by a vertex. Knowledge of angles is made use of in many spheres of everyday life. Engineers and architects use it for designing roads; buildings etc. artists use their knowledge of angles to sketch portraits and paintings.  One of such angles is a straight angle? What is it and how is it formed? Let us learn more about it.

Definition

A straight angle is an angle, whose vertex point has a value of 180 degrees. In other words, a straight angle is an angle that forms a straight line. The arms of a straight angle lie on the opposite side of the vertex. Straight angles are sometimes also referred to as flat angles.  It is important to note here that half of a straight angle is a right angle. Since a straight angle is 180 degrees, half of it will be 90 degrees which is a right angle.

The following is the description of a straight angle – 

Let us understand with an example.

Check whether the angles 125° and 53° can form a straight angle.

We have learned that a straight angle is formed when the sum of the angles is equal to 180o. Therefore, for the given two angles to form a pair of straight angles, their sum has to be equal to 180o. Hence, let us check the sum of the two given angles.

125° + 53° = 178o

We can see that the sum of the two given angles does not equal 180o

Hence, the two given angles, 125° and 53° cannot form a straight angle .

What changes could we have made in the other angle if we kept the first angle as 125° and the pair formed a straight angle? Let us check.

We already know that for a pair of angles to form a straight angle the sum of the angles is equal to 180o. Let the other angle be p. therefore, we have,

125° + p° = 180o

⇒ p° = 180o – 125°

⇒ p° = 55o

Hence, a pair of straight angles could have been formed if we increased the second angle by 2 o.

Degree

We now know that whenever two rays join together they form an angle and the angle subtended by two rays in opposite directions is called a straight angle. In degrees, the straight angle is represented as 180° and in radians; it is denoted with pi (π).

Properties of Straight angles

The following are the properties of straight angles

  1. The measure of a straight angle is always exactly half of a revolution. We know that the measure of one revolution is equal to360 o. therefore, half of the revolution, which is 180 o is the measure of a straight angle.
  2. A straight angle modifies the direction of a point.
  3. Two right angles can be joined together to form a straight angle.
  4. The arms of a straight angle are always extending in opposite directions.
  5. A straight angle is formed by rotating one ray by 180° with respect to another ray.
  6. When we move anticlockwise the straight angle measures + 180 0.
  7. When we move clockwise the straight angle measures – 180 0.

How to Draw a Straight angle using a Protractor?

Although a straight angle can be drawn by just drawing a straight line, there are at times, when you need to measure and locate straight angles, especially when it comes to complex diagrams. This is where drawing or locating a straight angle becomes easier and accurate when one uses a protractor. So, what are the steps that we follow to draw or locate a straight angle using a protractor? The following are the steps required – 

  1. The first steps towards drawing a straight angle using a protractor are to draw a straight-line OA, with an arrowhead at A. 
  2. Now we keep the protractor on this line such that the baseline of the protractor is aligned over OA. It is important to note here that the A should be pointing towards the 0°of the protractor. 
  3. In the next steps, we start from 0° and move towards 180° marking of the protractor. On arriving at 180°, we mark a point B on the paper. 
  4. In the last step we join the vertex of the line OA with the point B. The second ray OB will have the arrowhead at B. Thus we get our straight angle in the form of AOB.

Straight Angles in Real Life

There are many instances in real life where we make use of straight angles. Let us discuss some of them.

Clock

Have you ever noticed the different angles made by the hands of a clock at different times? Check the time shown by the clock below?

Can you notice the angle made by the hands of a clock when it reads 6:00 pm. They form a straight angle at 180o.

See Saw

Have you ever observed the angle made by a see saw when it is at a balanced position? Both the ends of a see saw make an angle of 180 o when they are in alignment with each other.  

Stairs

Observe the angle made by the base of the stairs below – 

What do you observe? The base of the stairs makes an angle of 180 o and hence forms a straight angle. 

Are supplementary angles Straight angles?

Two angles are said to be supplementary if their sum is 180 o. For example, two angles, 130 o and 50o are supplementary because their sum, 130 o + 50 o = 180 o. Together, the supplementary angles form a straight line. 

When the sum of two angles is 180 o, i.e. when two angles are supplementary angles, they are said to be supplements of each other. This means that the angles, 130 o and 50 o, since their sum is 180 o, therefore the angle 130 o is the supplement of 50 o and vice versa.

This means that a pair of supplementary angles can form a straight angle is they have a common arm. 

For example, let us consider the below figure

Here the line AB forms a straight line. Since the angle formed by a straight line is always equal to 180o, therefore, ∠ x + ∠ y = 180o

Hence, the pair of supplementary angles in the above example forms a straight angle.

Examples

Example 1 Find the value of ∠P  and  ∠Q if ∠P  and  ∠Q form a straight angle and 

∠P = 2x + 15 and   ∠Q = 5x – 38

Solution We have been given that ∠P  and  ∠Q form a straight angle. We have also been given that ∠P = 2x + 15 and   ∠Q = 5x – 38

Now, we know that if two angles form a straight angle, their sum will be equal to 180o. Now, since ∠P  and  ∠Q form a straight angle, therefore. 

∠ P + ∠ Q = 180o ………………………………… (1)

But, in this case we have not been given a constant value of any of the two angles. However, we have been provided with a relational values of both and P and Q in terms of x which has been stated as   ∠P = 2x + 15 and   ∠Q = 5x – 38

Therefore, in order to find the values of ∠P  and  ∠Q we will have to find the value of x. To get the value of x, first we should replace the values of ∠P  and  ∠Q  in the given equation  (1). Substituting these values we will get – 

2x + 15 + 5x – 38 = 180

⇒ 7x – 23 = 180

⇒ 7x = 180 + 23

⇒ 7x = 203

⇒ x = 2037 = 29

Therefore, x = 29o

Now, that we have the value of x, we can obtain the value of ∠P  and  ∠Q 

∠P   = 2x + 15

⇒ ∠P   = ( 2 x 29 )+ 15

⇒ ∠P   = 58 + 15

⇒ ∠P   = 73o

Also,

 ∠Q  = 5x – 38

 ∠Q  = ( 5 x 29 ) – 38

 ∠Q  = 145 – 38

  ∠Q  = 107o

Therefore, the values of the two angles forming a straight angle are P   = 73o and Q  = 107o

We can also check whether our answer is correct by substituting the values of ∠P  and  ∠Q in equation (1)

Substituting the values, we will get.

L.H.S 

∠ P + ∠ Q 

=73o + 107o

= 180o = R.H.S Hence our answer is correct.

Example 2 Calculate the value of θ in the figure below. Also, find the value of the three angles.

Solution From the given figure we can see that the given line is a straight line and three angles are formed on this straight line. Hence the sum of these three angles will be equal to 180o. Therefore,

5θ  + 4  + θ – 2 + 3θ + 7 = 180o

⇒ 9θ + 9 = 180o

⇒ 9θ = 180o – 9

⇒ 9θ = 171ο

⇒ θ = 171/9ο

θ = 19ο

Now that we know the value of θ we can find the value of the three angles. Substituting the value of θ in the three angles we will get,

First angle = 5θ  + 4   = ( 5  x 19 ) + 4 = 95 + 4 = 99o

Second angle =  θ – 2 = 19 – 2 = 17o

Third angle = 3θ + 7 = ( 3 x 19 ) + 7 = 57 + 7 = 64o

Hence, the three angles will be 99o, 17o, and  64o

Example 3 The ratio of a pair of angles forming a straight angle is 1 : 8. Find the two measures of the two angles?

Solution We have been given that the ratio of a pair of angles forming a straight angle is 1 : 8. Now, we also know that two angles are said to angles form a pair of straight angle if their sum is 180o. This means that the sum of these two angles should also be equal to 180o.

Let the two angles be represented by x and 8x.

This means that

x + 8x = 180o

  ⇒ 9x = 180o

⇒ x = 20o

Now that we know the value of x we can find out the value of the other two angles as well.

We have,

First angle = x  = 20o

Second angle = 8x = 8 x 20 = 160o

Hence, the pair of angles forming a straight angle will be 20o and 160o

Example 4 If angles of measures (x — 2)° and (2x + 5)° are a pair of angles forming a straight angle. Find the measure of the two angles.

Solution We have been given that the pair of angles that form a straight angle. These angles are – (x — 2)° and (2x + 5)°

Now, we know that two angles are said to form a straight angle if their sum is 180o. This means that the sum of these two angles should also be equal to 180o. In other words, since (x – 2)° and (2x + 5)° represent a pair of angles forming a straight angle, then their sum must be equal to 180°.

Therefore, we get the equation, (x – 2) + (2x + 5) = 180

We will solve this equation for x

x – 2 + 2x + 5 = 180

⇒ x + 2x – 2 + 5 = 180

⇒ 3x + 3 = 180

⇒ 3x = 180 — 3

⇒ 3x = 177

⇒ x = 177/3

⇒ x = 59°

Now, that we know the value of x we can find the value of the two angles.

First angle = x – 2 = 59 – 2 = 57°

Second angle = 2x + 5 = ( 2 × 59 ) + 5 = 118 + 5 = 123°

Therefore, the two angles that are forming a straight angle are 57° and 123°. 

Example 5 Find the value of “ a “ in the following figure – 

Solution We have been given that the pair of angles that form a straight angle. These angles are 63 o and “ a “ .

Now, we know that two angles are said to form a straight angle if their sum is 180o. This means that the sum of these two angles should also be equal to 180o. In other words, since 63 o and “ a “ represent a pair of angles forming a straight angle, then their sum must be equal to 180°.

Therefore, we get the equation,

63 o + a = 180 o

⇒ a = 180 o – 63 o 

⇒ a = 117 o

Hence, the value of angle a in the above figure is 117 o

Example 6 Find all the three angles in the given figure – 

Solution We have been given that a set of three angles that form a straight angle. These angles are x, 2 x and 3 x.

Now, we know that two angles are said to form a straight angle if their sum is 180o. This means that the sum of these two angles should also be equal to 180o. In other words, since x, 2 x and 3 x represent a set of angles forming a straight angle, then their sum must be equal to 180°.

Therefore, we get the equation,

x + 2 x  + 3 x = 180 o …………………….. ( 1 )

⇒ 6 x  = 180 o

⇒ x  = 180 / 6

⇒ x  = 30 o

Now, that we have got the value of x, we will find the value of all the three angles. We have,

First angle = x = 30 o

Second Angle = 2 x  = 2 x 30 o = 60 o

Third angle = 3 x  = 3 x 30 o = 90 o

Hence, the three angles in the given figure are 30 o, 60 o, and 90 o.

Key Facts and Summary

  1. A straight angle is an angle, whose vertex point has a value of 180 degrees.
  2. In degrees, the straight angle is represented as 180° and in radians; it is denoted with pi (π).
  3. The half of a straight angle is a right angle. Since a straight angle is 180 degrees, the half of it will be 90 degrees which are a right angle.
  4. The measure of a straight angle is always exactly half of a revolution. We know that the measure of one revolution is equal to 360 o. therefore, half of the revolution, which is 180 o is the measure of a straight angle.
  5. A straight angle modifies the direction of a point.
  6. Two right angles can be joined together to form a straight angle.
  7. The arms of a straight angle are always extending in opposite directions.
  8. A straight angle is formed by rotating one ray by 180° with respect to another ray.
  9. When we move anticlockwise the straight angle measures + 180 0.
  10. When we move clockwise the straight angle measures – 180 0.
  11. Two angles are said to be supplementary if their sum is 180 o.
  12. A pair of supplementary angles can form a straight angle if they have a common arm.