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