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

Question 18: Find the number of side of a regular polygon if it is given that the ratio of an interior angle and an exterior angle is .

Answer:

Given: Interior angle Exterior angle

Question 19: The measure of each interior angle of a regular polygon is . Determine the interior angle of another regular polygon the number of whose sides is twice of the first polygon.

Answer:

Let the number of sides of the first polygon

Therefore

For the second polygon:

Therefore Internal angle =

Question 20: Find the number of sides of the regular polygon if it is given that an interior angle and an exterior angle are in the ratio of .

Answer:

Given: Interior angle Exterior angle

Question 21: Show that the diagonals of a regular pentagon are equal.

Answer:

To prove:

Consider and

Since is a regular pentagon,

(given)

(given)

and (given)

Therefore (by S.A.S criterion)

Therefore (corresponding sides of congruent triangles are equal)

Hence the diagonals are equal from any vertex.

Question 22: The number of sides of two regular polygons are in the ratio of and their interior angles are in the ratio of , find the number of sides of the polygon.

Answer:

Polygon 1: Sides

Polygon 2: Sides

Given:

… … … … … i)

Also, their interior angles are in the ratio of

Therefore

… … … … … ii)

Substituting i) in ii) we get

Therefore

Question 23: The number of sides of two regular polygons are in the ratio and their sums of their interior angles are in the ratio . Find the number of sides of each polygon.

Answer:

Polygon 1: Sides

Polygon 2: Sides

Given:

… … … … … i)

Also, their sum interior angles are in the ratio of

Therefore

… … … … … ii)

Substituting i) in ii) we get

Therefore

Question 24: The difference between the exterior angles of two regular polygons is . If the number of sides of a polygonal is one more than the other, find the number of sides of each Polygon.

Answer:

Polygon 1: Sides

Polygon 2: Sides

Exterior angle of polygon 1

Exterior angle of polygon 2

Therefore

Now n cannot be a negative number. Hence . This implies that polygon 2 has sides.

Question 25: A heptagon has equal angles each of and three equal angles. Find the measure of equal angles.

Answer:

Heptagon:

Sum of interior angles

Therefore

Question 26: Find the number of sides of a polygon if the sum of interior angles is six times the sum of its exterior angles.

Answer:

Let the number of sides

Given sum of interior angles is six times the sum of its exterior angles

Therefore

Question 27: If the sum of interior angles of a pentagon are in the ratio , find the angles.

Answer:

Given:

Let the angles be , and

Sum of interior angles

Therefore

Therefore the angles are

Question 28: If the angles of a hexagon are and , find the value of .

Answer:

Angles of a hexagon are and

Number of sides:

Sum of interior angles

Therefore

Therefore the angles are and

Question 29: The angles of a pentagon are and Find the value of .

Answer:

Pentagon:

Sum of interior angles

Therefore

Question 30: The measures of three exterior angles of a hexagon are and , if each of the remaining exterior angles is , find the value of .

Answer:

Hexagon:

Sum of the exterior angles

Therefore

Question 31: In the adjoining figure is a regular pentagon. Find the measures of the angles marked .

Answer:

Regular pentagon:

Interior angle

In the quadrilateral

Since (sides of a regular pentagon)

Therefore

Therefore

Hence

Question 32: In a regular hexagon , prove that is an equilateral triangle.

Answer:

To prove: is equilateral triangle

Consider and

Therefore (By S.A.S criterion)

Similarly,

Therefore

is equilateral

Question 33: In a regular pentagon , show that is parallel to

Answer:

To prove:

Interior angle

Noe in

Similarly,

Therefore

Therefore since the sum of the interior alternate angles is

Question 34: If in a pentagon , we have

(i) and , find the value of .

Answer:

Regular pentagon:

Sum of interior angles

Therefore

(ii) and sides and when produced meet at right angles, find and

Answer:

Therefore

. Hence

(iii) and prove that and

Answer:

Consider and

Therefore (By S.A.S criterion)

Therefore

Therefore

Also since

Therefore

(iv) and are produced to meet at , prove that

Answer:

Assuming that is a regular pentagon

To prove:

Therefore

(v) and , find and

Answer:

Sum of internal angles

Therefore

Therefore

Question 35: is a regular pentagon such that diagonal divides into two parts. Find the ratio

Answer:

Regular pentagon:

Internal angle

In

Therefore

Therefore

Therefore the ratio of