An angle results when two lines meet at a point, and the size of the 'opening' between the two lines is referred to as an angle and denoted by the symbol ∠. Two angles are said to be supplementary if their measure adds up to 180 degrees. Angles are measured using degrees and radians, which stand for circularity or rotation. The knowledge of angles is vital for solving geometry questions, hence it is an important subject in US competitive exams such as the SAT and ACT. Engineers and architects utilize these principles to construct roads, buildings, and sports centers. The mastery of various kinds of angles like acute, obtuse, right, reflex, and straight angles can help you perform well while answering geometry questions.

What are Supplementary Angles?

Supplementary angles refer to a pair of angles that always add up to 180°. When two lines intersect each other we get 4 pairs of supplementary angles. When supplementary angles are combined, they make a straight angle (180 degrees). In other words, if, \( Angle 1 + Angle 2 = \angle 180^{\circ} \) angles 1 and 2 are supplementary. Angles 1 and 2 are referred to as ‘supplements’.

Supplementary Angles Example 1:

In between the parallel lines, supplementary angles are formed on the same side of the transversal. 3 and 5, as well as 4 and 6, are supplementary angles and 1,5 are Corresponding angles.

Supplementary Angles Example 2:

Opposite angles in a parallelogram have equal lengths, each parallelogram contains two pairs of equal angles. The angles in a parallelogram add up to 360^{\circ} because there are two pairs of angles, one on each side, and adjacent angles are supplementary (their sum equals \(180^{\circ}\) ).

Types of Supplementary Angles

There are 2 types of angles that are supplementary:

1. Adjacent Supplementary Angles
2. Non-Adjacent Supplementary Angles

Adjacent Supplementary Angles

Adjacent supplementary angles are two supplementary angles that share a common vertex and an arm. They are formed by the intersection of two lines.

\( \angle COB \) and \( \angle AOB \) are adjacent supplementary angles because they share a common vertex, O, and an arm, OB.

They also add up to 180 degrees: \( \angle COB+\angle AOB=70^{\circ}+110^{\circ}=180^{\circ} \)

As a result, these two angles are supplementary angles that are adjacent.

Non-Adjacent Supplementary Angles

Non-adjacent supplementary angles are two supplementary angles that are NOT adjacent or contiguous to each other.

\( \angle ABC \) and \( \angle PQR \) are non-adjacent angles since they share neither a vertex nor an arm. They also \(sum to 180^{\circ}\), as \( \angle ABC+\angle PQR=79^{\circ}+101^{\circ}=180^{\circ}\)

As a result, these are non-adjacent supplementary angles. When non-adjacent supplementary angles are added together, they make a straight angle.

How to Find Supplementary Angles?

Supplementary angles can be calculated by subtracting the given angle from 180 degrees. Use the following formula to find the other angle:

\( \angle x = 180^{\circ}-\angle y \)

where angles x and y are given angles

Supplementary Angles Theorem

Supplementary Angles Theorem Statement: If two angles are supplementary to the same angle, the two angles are said to be congruent.

Supplementary Angles Theorem Proof:

\( \angle AOP+\angle POB = 180^{\circ} \)

\( \Rightarrow \angle AOP=180^{\circ}-\angle POB \) ………(1)

\( \angle POB + \angle QOB=180^{\circ} \)

\( \Rightarrow \angle QOB = 180^{\circ}-\angle POB \) …..(2)

From equation (1) and equation(2)

\( \angle QOB=\angle AOP \)

Properties of Supplementary Angles

Some of the fundamental properties of supplementary angles are as follows:

• When two angles add up to 180 degrees, they are said to be supplementary.
• Any two intersecting lines make 4 pairs of supplementary angles
•  Supplementary angles can be adjacent or nonadjacent to one another like vertical angles.

Difference between Complementary and Supplementary Angles

Complementary Angles	Supplementary Angles
When two angles sum up to 90 degrees, they are said to be complimentary.	When two angles sum up to 180 degrees, they are called supplementary.
They form a right angle together	They form a straight angle together
They are formed when a line segment divides a right angle	They are formed when a line segment intersects another line segment.
Example of Complementary Angle	Example of Supplementary Angle

Solved Examples of Supplementary Angles

The supplementary angle ideas learned are used to solve the following supplementary angle examples.

Example 1: Determine the values of Angles A and B if they are supplementary angles with \( \angle A=\left ( 2x+10 \right )^{\circ} \) and \( \angle B=\left ( 6x-46 \right )^{\circ} \)

Solution: Because Angles A and B are supplementary, their sum equals 180°.

\( \angle A +\angle B = 180^{\circ} \)

\( \left ( 2x+10 \right )+\left ( 6x-46 \right ) = 180^{\circ} \)

\(8x-36=180\)

\(8x=216\)

\(x=27\)

As a result angle \( A=2\left ( 27 \right )+10=64^{\circ} \) and \( B=6\left ( 27 \right )-46= 116^{\circ} \)

Example 2: Determine the angle that is 68 degrees less than its complement.

Solution: We may denote the desired angle with x, hence the complementary angle is \(\left (90^{\circ}-x\right )\) The difference between the angle and its complement is equal to \(38^{\circ}\) , according to the query. As a result, we have:

\(\left ( 90-x \right )-x=38\)

\(90-2x=38\)

\(-2x = 38-90\)

\(-2x = -38\)

\(x = 19\)

the angle is \(19^{\circ}\)

Example 3: Determine the angle QPS in the cyclic quadrilateral given below.

Solution: \( \angle QPS \) is the opposite angle of \( \angle SRQ \)

The inscribed quadrilateral theorem states that

\( \angle QPS + \angle SRQ = 180^{\circ} \) (Supplementary Angles)

\( \angle QPS + 60^{\circ} = 180^{\circ} \)

Now subtract \( 60^{\circ} \) both sides

\( \angle QPS= 120^{\circ} \)

As a result, the angle \( \angle QPS \) is \( 120^{\circ} \)

Example 4: Two parallel lines are cut by a transversal in the illustration below. Which of the following highlighted angles is supplementary to \( \angle 1 \)?

Solution: The supplementary angle to \( \angle 1 \) is \( \angle 6 \)

\( \angle 1 \) is an obtuse angle, therefore any one acute angle combined with any obtuse angle is a supplementary angle. This is the only sharp angle marked.

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