Chapter 5, Problem 19RP

a.

To determine

To Find: Whether the statement is always true, if sometime is true or never true?

Answer to Problem 19RP

The given statement is always true, hence this statement is written as A

Explanation of Solution

Given:

If the diagonals of a quadrilateral are congruent, the quadrilateral is an isosceles trapezoid.

Concept Used:

An isosceles trapezoid is a convex quadrilateral having a line of symmetry bisecting one pair of opposite sides.

Calculation:

If the diagonals of a quadrilateral are congruent, the quadrilateral is an isosceles trapezoid. This statement is always true, hence this statement is written as A

Conclusion:

The given statement is always true, hence this statement is written as A

b.

To determine

To Find: Whether the statement is always true, if sometime is true or never true?

Answer to Problem 19RP

The given statement is always true, hence this statement is written as A

Explanation of Solution

Given:

If the diagonals of a quadrilateral divide each angle into two 45 degrees angles, the quadrilateral is a square.

Concept Used:

Square is a quadrilateral whose all sides and all angles are of equal measure. Also diagonal divides the angles at each corner into 45 degrees.

Calculation:

If the diagonals of a quadrilateral divide each angle into two 45 degrees angles, the quadrilateral is a square. This statement is always true, hence this statement is written as A

Conclusion:

The given statement is always true, hence this statement is written as A

c.

To determine

To Find: Whether the statement is always true, if sometime is true or never true?

Answer to Problem 19RP

The given statement is sometime true, hence this statement is written as S.

Explanation of Solution

Given:

If a parallelogram is equilateral, it is equiangular.

Concept Used:

Equilateral parallelogram is a parallelogram of which all sides are equal. Rhombus is of this category but it is not equiangular, while square is equilateral parallelogram which is equiangular also.

Calculation:

If a parallelogram is equilateral, it is equiangular. This statement is sometimes true, since Rhombus is of this category but it is not equiangular, while square is equilateral parallelogram which is equiangular also. Hence this statement is written as S.

Conclusion:

The given statement is sometime true, hence this statement is written as S.

d.

To determine

To Find: Whether the statement is always true, if sometime is true or never true?

Answer to Problem 19RP

The given statement is always true, hence this statement is written as A.

Explanation of Solution

Given:

If two of the angles of a trapezoid are congruent, the trapezoid is isosceles.

Concept Used:

A trapezoid is a quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a convex quadrilateral having a line of symmetry bisecting one pair of opposite sides.

Calculation:

If two of the angles of a trapezoid are congruent, the trapezoid is isosceles. This statement is always true. Hence this statement is written as A.

Conclusion:

The given statement is always true, hence this statement is written as A.