Supplementary and Complementary Angles

Complementary and supplementary angles are the two different sorts of angles. A complementary angle is one that measures 90° whereas a supplementary angle is one that measures 180°. 

• An angle is created when two lines share a vertex. 
• The vertex, arms, interior, and exterior angles are some of the components of an angle. 
• At the place where two lines join, four angles are created. An angle is denoted by the symbol ∠. 

Key terms: Supplementary, Complementary, Angles, Vertex, Arms, Exterior, Interior, Adjacent, Non-Adjacent.

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Supplementary Angles 

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Angles with a supplementary sum are those whose sum is 180 degrees. For instance, the total of ∠120° and ∠60° is 180°, hence they are supplementary angles. In the same way, complementary angles sum to 90 degrees.

 

• A straight line and a straight angle are formed when the two additional angles are combined.
• It should be emphasized, however, that the two angles that complement one another do not necessarily have to be close to one another. 
• Any two angles can therefore be supplementary angles if their total is 180 degrees.

Supplementary Angles

Some Examples of Supplementary Angles

Examples of supplementary angles include the following:

• ∠130° + ∠50° = ∠180°
• ∠90° + ∠90° = ∠180°
• ∠150° + ∠40° = ∠180°
• ∠98° + ∠82° = ∠180°

Properties of Supplementary Angles

Here are some important properties of supplementary angles include : 

• The following are crucial characteristics of additional angles:
• When the sum of the two angles is 180 degrees, they are referred to as supplementary angles.
• For a line to be straight, the two angles do not need to be close together.
• The "Straight" line is represented by "S" in supplementary angles. They thus create 180 degrees.

Adjacent and Non-Adjacent Supplementary Angles

Types of Supplementary Angles are adjacent & non-adjacent:

Adjacent Supplementary angles

Adjacent supplementary angles are supplementary angles with a common arm and a common vertex. The same line segment and vertex are shared by the adjacent additional angles.

Adjacent supplementary angles

Non-adjacent Supplementary angles

Non-adjacent supplementary angles are those supplementary angles that do not share a common arm or a shared vertex. The line segment or vertex are not shared by the non-adjacent additional angles.

How to Find Supplementary Angles?

If one angle is x, then the second angle is 180° - x since the two angles together make a linear angle. Here, linearity demonstrates that the angles' characteristics stay constant. Consider several instances of trigonometric ratios, as;

Sin (180° – A) = Sin A

Cos (180° – A) = – Cos A (new quadrant)

Tan (180° – A) = – Tan A

Read More: Differential Equation 

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Complementary Angles

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Angles that are complementary are those whose sum angle is less than 90°. In other words, if two angles combine to form a right angle, they are said to be complementary. 

Complementary Angles

Properties of Complementary Angles

Properties of complementary angles are: 

• Two Angles are considered to be Complementary Angles in accordance with the concept of Supplementary Angles if the total of their measurements is ∠90°.
• A supplementary Angle need not be on the same line; it can be on another line as long as it measures ∠90°.

How to Find Complementary Angles?

Consider the case when one angle is x and the other is 90° - x. Therefore, these complementary angles are used for trigonometric ratios when one ratio is 90 degrees complementary to another ratio.

Sin (90°- A) = cos A and cos (90°- A) = sin A

Tan (90°- A) = cot A and cot (90°- A) = tan A

Sec A (90°- A) = cosec A and cosec (90°- A) = sec A

Read More: Trapezoid Formula

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Solved Examples

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Ques. Two more angles are in the proportion of 7: 8. Determine the angles' measurements.

Solution: Let the ratio be x. 

If one angle is 7x, then the other angle is 8x. 

Therefore, 7x + 8x = 180° 

15x = 180

x = 180/15

x = 12

Put the value of x = 12

One angle is 7x 

= 7 × 12

= 84° 

And the other angle is 8x 

= 8 × 12

= 96° 

Therefore, the two supplementary angles are 84° and 96°. 

Ques. Check to see if 115° and 65° are two supplementary angles.

Solution: 115° + 65° = 180°

Hence, they are a pair of supplementary angles.

Ques. Find the complementary of the 2/3 of a 90° angle.

Solution: Convert 2/3 of 90°

2/3 × 90° = 60°

Complement of 60° = 90° - 60° = 30°

Therefore, complement of the angle 2/3 of 90° = 30°

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Things to Remember 

• Complementary Angles are adjacent or on the same line with a 90-degree sum.
• Supplementary Angles are angles that sum up to 180 degrees, regardless of their proximity.
• Two adjacent Angles have a common side, arm, or vortex, indicating they are side by side.
• Angles that are obtuse have a greater than 90-degree angle but a lesser than 180-degree angle. So, an obtuse angle might be between 90⁰ and 180⁰ in value.
• A right Angle is one that is precisely 90 degrees in length. 
• A triangle is referred to as a right-angled triangle if this right angle is present. 
• Every angle in a square or rectangle is a right angle, or one that measures 90 degrees.
• Reflex Angles are those that are larger than 180 degrees but less than 360 degrees in length. 
• Reflex Angle values thus range from 180 degrees to 360 degrees.

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