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

Answer:

Given angles:

The sum of the interior angles of a quadrilateral

Therefore the 4th angle

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

Answer:

The sum of the interior angles of a quadrilateral

Hence the angles are:

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

Answer:

… … … … … 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 (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:

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)

ii)

iii)

iv)

v)

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

ii) Sum of all interior angles of a polygon

iii) Sum of all interior angles of a polygon

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

Answer:

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:

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:

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

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

Answer:

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

Answer:

Since

In a quadrilateral sum of opposite angles

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:

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

ii) Given:

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

iii) Given:

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)

We know

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

ii)

We know

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

iii)

We know

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)

We know that sum of interior angles of a polygon

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

ii)

We know that sum of interior angles of a polygon

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

iii)

We know that sum of interior angles of a polygon

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: Exterior angle

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

Answer:

Given: 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

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:

Exterior angle

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

Answer:

Consider

Since is a regular pentagon,

(given)

(given)

and (given)

(by S.A.S criterion)

(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

… … … … … i)

Also, their interior angles are in the ratio of

… … … … … ii)

Substituting i) in ii) we get

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

… … … … … i)

Also, their sum interior angles are in the ratio of

… … … … … ii)

Substituting i) in ii) we get

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

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:

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:

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

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

Answer:

Therefore the angles are

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

Answer:

Angles of a hexagon are

Number of sides:

Therefore the angles are

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

Answer:

Pentagon:

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

Answer:

Hexagon:

Sum of the exterior angles

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

Answer:

In the quadrilateral

Since (sides of a regular pentagon)

Hence

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

Answer:

is equilateral triangle

Consider

(By S.A.S criterion)

Similarly,

is equilateral

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

Answer:

Noe in

Similarly,

since the sum of the interior alternate angles is

Question 34: If in a pentagon , we have

(i) , find the value of .

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

(iii) prove that

(iv) are produced to meet at , prove that

(v) , find

Answer:

(i) , find the value of .

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

. Hence

(iii) prove that

Consider

(By S.A.S criterion)

Also since

(iv) are produced to meet at , prove that

Assuming that is a regular pentagon

(v) , find

Sum of internal angles

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

Answer:

Internal angle

In

Therefore the ratio of