Question 1: Three angles of a quadrilateral are respectively equal to its fourth , and . Find angle.

Answer:

Number of sides

Given angles: and

The sum of the interior angles of a quadrilateral

Therefore the 4th angle

Question 2: In a quadrilateral , the angles and are in the ratio . Find the measure of each angle of the quadrilateral.

Answer:

Given:

Number of sides

The sum of the interior angles of a quadrilateral

Therefore

Hence the angles are: and

Question 3: ln a quadrilateral and. are the bisectors of and respectively. Prove that

Answer:

To prove:

… … … … … i)

Substituting in i)

Question 4: Prove that the sum of all the interior angles of a pentagon is .

Answer:

The pentagon comprises of three triangles and (please refer to the adjoining diagram)

Sum of the internal angles of

Question 5: What is the measure of each angle of a regular octagon?

Answer:

Regular Octagon

Interior Angle

Exterior Angle

Question 6: Find the number of sides of a regular polygon, when each of its angles has a measure of i) ii) iii) iv) v)

Answer:

i) Let the number of sides

We know Interior Angle

Given interior angle

Therefore

ii) Let the number of sides

We know Interior Angle

Given interior angle

Therefore

iii) Let the number of sides

We know Interior Angle

Given interior angle

Therefore

iv) Let the number of sides

We know Interior Angle

Given interior angle

Therefore

v) Let the number of sides

We know Interior Angle

Given interior angle

Therefore

Question 7: Find the number of sides of a polygon the sum of whose interior angles is i) ii) right angles iii) straight angles

Answer:

i) Sum of all interior angles of a polygon

Therefore

ii) Sum of all interior angles of a polygon

Therefore

iii) Sum of all interior angles of a polygon

Therefore

Question 8: Find the number of degrees in each exterior angle of regular pentagon.

Answer:

Regular pentagon:

Exterior angle of a regular polygon of sides

Therefore exterior angle of pentagon

Question 9: The measure of angles of hexagon are . Find the value of .

Answer:

Regular hexagon:

Sum of interior angles

Therefore

Hence the Exterior angle

Question 10: In a convex hexagon, prove that the sum of all interior angles is equal to twice the sum of exterior angles formed by producing the sides in the same order.

Answer:

Convex hexagon:

Sum of interior angles

Sum of exterior angles of a hexagon

Hence Sum of all interior angles sum of all exterior angles

Question 11: The sum of the interior angles of a polygon is three times the sum of its exterior angles. Determine the number of sides of the polygon.

Answer:

Given: Sum of all interior angles sum of all exterior angles

Therefore

Question 12: Determine the number of sides of a polygon whose exterior and interior angles are in the ratio of .

Answer:

Interior Angle

Exterior angle

Given:

Question 13: is a regular hexagon. Determine each angle of

Answer:

Regular hexagon:

Therefore Interior Angle

Since

In a quadrilateral sum of opposite angles

Therefore

Therefore

Question 14: Is it possible to construct a regular polygon the measure of whose each interior angle is i) ii) and iii)

Answer:

i) Given: Interior angle

Interior Angle

Therefore

Therefore it is possible to construct a regular polygon with an interior angle of

ii) Given: Interior angle

Interior Angle

Therefore

Therefore it is not possible to construct a regular polygon with an interior angle of since should be a positive integer.

iii) Given: Interior angle

Interior Angle

Therefore

Therefore it is possible to construct a regular polygon with an interior angle of

Question 15: Is it possible to construct a regular polygon the measure of whose each exterior angle is i) ii) iii)

Answer:

i) Exterior angle

We know exterior angle

Therefore

Therefore it is possible to construct a regular polygon with an exterior angle of

ii) Exterior angle

We know exterior angle

Therefore

Therefore it is not possible to construct a regular polygon with an exterior angle of

iii) Exterior angle

We know exterior angle

Therefore

Therefore it is not possible to construct a regular polygon with an exterior angle of

Question 16: Can a regular polygon be described the sum of whose interior angles is i) ii) iii) right angles

Answer:

i) Sum of interior angles

We know that sum of interior angles of a polygon

Therefore

Hence it is not possible to construct a polygon where the sum of interior angles is

ii) Sum of interior angles

We know that sum of interior angles of a polygon

Therefore

Hence it is possible to construct a polygon where the sum of interior angles is

iii) Sum of interior angles

We know that sum of interior angles of a polygon

Therefore

Hence it is not possible to construct a polygon where the sum of interior angles is

Question 17: Determine the number of sides of regular, polygon the measure of whose each interior angle is double that of the exterior angle

Answer:

Given: Interior angle Exterior angle