CLASSIFICATION OF PAIR OF ANGLES

We shall classify a pair of angles in the following way for better understanding and usages.

(i) Two angles that have the exact same measure are called congruent angles.

(ii) Two angles that have their measures adding to 90° are called complementary angles.

(iii) Two angles that have their measures adding to 180° are called supplementary angles.

(iv) Two angles between 0° and 360° are conjugate, if their sum equals 360°.

Example 1 :

In the figure shown below,  let the lines l₁ and l₂ be parallel and t is transversal. Find the value of x.

Solution : 

From the given figure, 

∠(2x + 20)° and ∠(3x - 10)° are corresponding angles. 

So, they are congruent. 

Then, 

2x + 20  =  3x - 10

30  =  x

Example 2 :

In the figure shown below, let the lines l₁ and l₂ be parallel and m is transversal. Identify the pairs of angles which are congruent. 

Example 3 :

Find the value of x :

Solution :

From the picture above, it is clear that the angles (x+1), (x-1) and (x+3) are complementary.  

Then,

(x+1) + (x-1) + (x+3)  =  90

x + 1 + x - 1 + x + 3  =  90

Simplify.

3x + 3  =  90

Subtract 3 from each side. 

3x  =  87

Divide each side by 3.

x  =  29

So, the value of x is 29.

Example 4 :

Find the value of x :

Solution :

From the picture above, it is clear (5x+4), (x-2) and (3x+7) are supplementary angles. 

Then, 

(5x+4) + (x-2) + (3x+7)  =  180

5x + 4 + x -2 + 3x + 7  =  180

Simplify. 

9x + 9  =  180

Subtract 9 from each side.

9x  =  171

Divide each side by 9.

x  =  19

So, the value of x is 19.

Example 5 :

Find the conjugate of the angle measure 97°. 

Solution : 

Let x° be the conjugate of the angle measure 97°. 

Because x° and 97° are conjugate, their sum equals 360°.

Then, 

x° + 97°  =  360°

Subtract 97° from each side. 

x°  =  263°

So, the conjugate of the angle measure 97° is 263°. 

Example 6 :

If (x + 30)° and (2x - 60)° are conjugate, find the value of x.  

Solution : 

Because (x + 30)° and (2x - 60)° are conjugate, their sum equals 360°.

Then, 

(x + 30)° + (2x - 60)°  =  360°

x + 30 + 2x - 60  =  360

Simplify.

3x - 30  =  360

Add 30 to each side. 

3x  =  390

Divide each side by 3.

x  =  130

So, the value of x is 130.

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