Chapter 7.4, Problem 13PSB

a

To determine

The given statement “If the number of sides of an equiangular polygon is doubled, the measure of each exterior angle is halved” is always, sometimes or never be true.

Answer to Problem 13PSB

Always

, If the number of sides of an equiangular polygon is doubled, the measure of each exterior angle is halved

Explanation of Solution

Given information:

The number of sides of an equiangular polygon is doubled

The measure of exterior angle in a polygon of ‘ n’ sides is given by $\frac{360}{n}$

If sides are doubled, then obviously measure of exterior angle will get halved

b

To determine

The following statement “The measure of an exterior angle in decagon is greater than the measure of angle in quadrilateral” is A, S or N.

Answer to Problem 13PSB

Sometimes, the measure of an exterior angle in decagon is greater than the measure of angle in quadrilateral

Explanation of Solution

Given information:

The measure of an exterior angle in decagon

Since, the measure of an exterior angle in decagon is 36 $°$ and that of quadrilateral is 90 $°$ but sum of all the angles in both the polygons will never get changed

Thus, sometimes, the measure of an exterior angle in decagon is greater than the measure of angle in quadrilateral

c

To determine

“The regular polygon is equilateral” is always, sometimes or never be true.

Answer to Problem 13PSB

Always, the regular polygon is equilateral

Explanation of Solution

Given information:

The regular polygon

Since, all the sides of the regular polygon are equal in measure then all its sides will be equal

Thus, always the regular polygon is equilateral

d

To determine

To check:The given statement “An equilateral polygon is regular” is always, sometimes or never be true.

Answer to Problem 13PSB

Sometimes,an equilateral polygon is regular

Explanation of Solution

Given information:

The equilateral polygon

For a regular polygon, all sides have the same length and all interior angles are same.

Also, the property that in a regular polygon, all vertices lie on a circle.

Thus, Sometimes, an equilateral polygon is regular

e

To determine

To check: “If the midpoints of a scalene quadrilateral are joined in order, figure formed is equilateral” is always, sometimes or never be true.

Answer to Problem 13PSB

Sometimes, If the midpoints of a scalene quadrilateral are joined in order, figure formed is equilateral

Explanation of Solution

Given information:

The scalene quadrilateral

If the midpoints of the sides of a scalene quadrilateral are joined in order, the figure formed is equilateral.

An equilateral polygon is regular. 

If the midpoint of the sides of a rhombus are joined in order, the figure formed is equilateral but not equiangular.

Thus, sometimes, if the midpoints of a scalene quadrilateral are joined in order, figure formed is equilateral

f

To determine

To check: The given statement “If the midpoints of a rhombus are joined in order, figure formed is equilateral but not equiangular” falls under A, S or N.

Answer to Problem 13PSB

Never, if the midpoints of a rhombus are joined in order, figure formed is equilateral but not equiangular

Explanation of Solution

Given information:

The scalene quadrilateral

 If the midpoints of the sides of a rhombus are joined in order, the figure formed is equilateral but not equiangular

 If one of the angles of an isosceles triangle is 60°, the triangle is equilateral.

Thus, never, if the midpoints of a rhombus are joined in order, figure formed is equilateral but not equiangular