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.

Construction of Angles

In construction of angles here we are going to learn on how to construct the different angles by using of scale, pencil and compass. We can construct the angles 60°, 120°, 30°, 90°, 45°, 135°, 75°, 105° etc. by using the compass. Let’s learn them.

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10 Math Problems officially announces the release of Quick Math Solver and 10 Math Problems, Apps on Google Play Store for students around the world.

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Construction of an angle 60°

Steps:

1.    Draw a line segment AB.

2.    Draw an arc of any size with the centre at the point B, so that it cuts AB at point D.

3.    Cut the same size of the previous arc at E from the point D.

4.    Join the points E and B to meet C as shown in the figure.

5.    Then ∠ABC = 60°.

Construction of an angle 120°

Steps:

1.    Draw a line segment AB.

2.    Draw an arc of any size with the centre at B which cuts AB at C.

3.    Cut the same size of the previous arc at the point D from the point C.

4.    Again cut the same arc at E from the point D.

5.    Join the points E and B to meet F as shown in the figure.

6.    Then, ∠ABF = 120°.

Construction of an angle 30°

Steps:

1.    Construct an angle of 60° as before.

2.    Draw an arc from the point D and E which intersect at the point F as shown in the figure.

3.    Join the points B and F. Then BF bisects the ∠ABC.

4.    Then, ∠ABF = 30°.

Construction of an angle 90°

Steps:

1.    Construct the angles 60° and 120° as before.

2.    Draw same arcs from C and D so that the two arcs intersect at E as shown in the figure.

3.    Join the points O and E.

4.    Then, ∠AOE = 90°.

Construction of an angle 45°

Steps:

1.    Construct an angle of 90° as above.

2.    Draw same arcs from B and E so that they intersect at F.

3.    Join the points F and O as shown in the figure.

4.    Then, ∠FOA = 45°.

Construction of an angle 135°

Steps:

1.    Construct an angle of 90° as above.

2.    Draw same arcs from F and E so that they inersect at G.

3.    Join the points G and O as shown in the figure.

4.    Then, ∠AOG = 135°.

Construction of an angle 75°

Steps:

1.    Construct an angle 90° as before.

2.    Draw same arcs from D and F so that they intersect at the point G.

3.    Join the points G and B as shown in the figure.

4.    Then, ∠ABG = 90°.

Construction of an angle 105°

Steps:

1.    Construct an angle of 90° as before.

2.    Draw same arcs from U and T so that they intersect at the point V.

3.    Join the points Q and V as shown in the figure.

4.    Then, ∠AQV = 105°.

You can comment your questions or problems regarding the construction of angles by using compass here.