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