Pair of Angles

Pair of Angles

There are some pair of angles having some specific properties. Adjacent angles, linear pair of angles, right pair of angles, complementary angles, supplementary angles and vertically opposite angles are some special pair of angles. let’s learn them.

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Adjacent Angles

Two angles are said to be adjacent angles if they have a common arm and common vertex.

In the given figure, ∠AOC and ∠BOC are adjacent angles as they have a common arm OC and the common vertex O.

Adjacent angles never overlap with each other.

Linear Pair of Angles

If the sum of a pair of adjacent angles is 180°, the pair is called a linear pair of angles.

In the given figure, ∠AOC + ∠COB = 180°. So ∠AOC and ∠COB are linear pair of angles.

Right Pair of Angles

If the sum of a pair of adjacent angles is 90°, the pair is called a right pair of angles.

In the given figure, ∠AOC + ∠COB = 90°. So ∠AOC and ∠COB are right pair of angles.

Complementary Angles

Two angles are said to be complementary angles of each other if their sum is 90°.

In the given figure, ∠ABC = 35° and ∠DBC = 55° and ∠ABC + ∠DBC = 35° + 55° = 90°. ∴ ∠ABC and ∠DBC are complementary angles.

Similarly, ∠PQR = 70° and ∠STU = 20° and ∠PQR + ∠STU = 70° + 20° = 90°. ∴ ∠PQR and ∠STU are complementary angles.

Each of the angle is complement of the other.

Supplementary Angles

A pair of angles whose sum is 180° are called supplementary angles.

In the given figure, ∠AOC = 105° and ∠BOC = 75° and ∠AOC + ∠BOC = 105° + 75° = 180°. ∴ ∠AOC and ∠BOC are supplementary angles.

Similarly, ∠ABC = 35° and ∠DEF = 145° and ∠ABC + ∠DEF = 35° + 145° = 180°. ∴ ∠ABC and ∠DEF are supplementary angles.

Each of the angle is supplement of the other.

Vertically Opposite Angles [V.O.A.]

When two straight line segment intersect, the angles formed opposite to each other are called vertically opposite angles.

In the given figure, ∠PON and ∠QOM, ∠POM and ∠NOQ are vertically opposite angles.

Similarly, ∠TOU and ∠SOV, ∠UOS and ∠VOT are also vertically opposite angles.

Workout Examples

Example 1: Find the value of x in the given figure.

Solution: From the figure,

                          x + 75° = 180° -----------------> Linear pair of angles.

                or,     x = 180° – 75°

                or,     x = 105°

Example 2: Find the value of x from the given figure.

Solution: From the figure,

                          x + x + 40° = 90° -----------------> Right pair of angles.

                or,     2x = 90° – 40°

                or,     2x = 50°

                or,      x = 50°/2

                or,     x = 25°

Example 3: Find the values of a, b and c in the given figure.

Solution: From the figure,

                          a + 40° = 180° -----------------> Linear pair of angles.

                or,     a  = 180° – 40°

                or,     a = 140°

                         b = 40° ------------------> Vertically opposite angles.

                         c = a ------------------> Vertically opposite angles.

                            = 140°

                ∴  a = 140°, b = 40° and c = 140°

Example 4: Find the values of x, a and b in the given figure.

Solution: From the figure,

                         2x + 3x = 180° -----------------> Linear pair of angles.

                or,     5x  = 180°

                or,     x = 180°/5

                or,     x = 36°

                         a = 3x ------------------> Vertically opposite angles.

                            = 3×36°

                            = 108°

                         b = 2x ------------------> Vertically opposite angles.

                            = 2×36°

                            = 72°

                ∴  x = 36°, a = 108° and b = 72°

Example 5: Find the values of x, a and b in the given figure.

Solution: From the figure,

                          2x – 30° = x + 20° -----------------> Vertically opposite angles.

                or,     2x – x = 20° + 30°

                or,     x = 50°

                         a + x + 20° = 180° ------------------> Linear pair of angles.

                or,     a + 50° + 20° = 180°

                or,     a + 70° = 180°

                or,     a = 180° – 70°

                or,     a = 110°

                         b = a ------------------> Vertically opposite angles.

                            = 110°

                ∴  x = 50°, a = 110° and b = 110°

You can comment your questions or problems regarding the pair of angles here.