Pairs of Angles (Complementary, Supplementary, Adjacent, Linear Pair)

1. Angle Pairs – Introduction

Angle pairs are formed when two angles have a special relationship with each other. These relationships help us understand how angles behave when lines meet or when two angles lie next to each other. Learning these pairs makes it easier to solve angle-related problems and read diagrams accurately.

2. Complementary and Supplementary Angles

Two angles can work together to form bigger or smaller total angles. This gives rise to two important angle pairs: complementary and supplementary angles. Their definitions depend on the sum of the two angles.

2.1. Complementary Angles

Complementary angles are two angles whose measures add up to \( 90^\circ \). They do not have to be next to each other; only their angle measures must make \( 90^\circ \) in total.

Example: If one angle is \( 30^\circ \), the other must be \( 60^\circ \) to make a complement.

  \) 30°     +     \) 60°  =  90°

2.2. Supplementary Angles

Supplementary angles are two angles whose measures add up to \( 180^\circ \). Like complementary angles, they do not need to be adjacent, but their total must be exactly \( 180^\circ \).

Example: Angles measuring \( 110^\circ \) and \( 70^\circ \) are supplementary because their sum is \( 180^\circ \).

  \) 110°     +     \) 70°  =  180°

3. Adjacent Angles and Linear Pair

Sometimes two angles share a common side and a common vertex. These angles sit next to each other and form a special arrangement. When the two angles also lie on a straight line, they form a linear pair.

3.1. Adjacent Angles

Adjacent angles are two angles that:

share the same vertex,
share one common arm,
and lie on the same side of the common arm.

They sit next to each other like two pieces placed side by side.

   A
    \)
     \)  (Two angles with a common arm)
      C

3.2. Linear Pair of Angles

A linear pair is formed when two adjacent angles lie on a straight line. This means their non-common arms form a straight angle, or \( 180^\circ \).

In a linear pair, the two angles are always supplementary because their sum must be \( 180^\circ \).

 -------+-------
        |
        |

4. Vertically Opposite Angles

When two lines intersect, they form two pairs of angles that face each other. These are called vertically opposite angles. They are always equal in measure because the lines create symmetric opposite openings.

For example, if intersecting lines form one angle of \( 50^\circ \), the angle directly opposite it will also be \( 50^\circ \).

   \        /
    \  50° /
     X-----
    / 50°  \
   /        \

5. Real-Life Situations of Angle Pairs

Angle pairs occur naturally in everyday settings:

Complementary: two hands of a clock when the small angle is 90°.
Supplementary: angles formed by a straight road branch.
Adjacent: corners of a window frame where two edges meet.
Linear pair: the angles formed when a pole stands on level ground.
Vertically opposite: crossing roads forming an X-junction.

Seeing these real examples makes angle relationships easier to remember.

Maths