Chapter 5.6, Problem 1PSA

(a)

To determine

To state: The property that proves that the quadrilateralQUADis a parallelogram.

Answer to Problem 1PSA

QUADis a parallelogram because its diagonals bisect each other.

Explanation of Solution

Given:The diagonalsAQand UDof a quadrilateral QUADbisects each other.

QUADis a parallelogram because its diagonals bisect each other.

(b)

To determine

To state: The property that proves that the quadrilateral QUADwith it’s a pair of opposite sides equal and parallel, is a parallelogram.

Answer to Problem 1PSA

A pair of opposite sides in the given quadrilateral QUADis congruent and parallel so it is a parallelogram.

Explanation of Solution

Given:In quadrilateral QUAD, $UQ\cong AD$ and $UQ\parallel AD$ .

Here a pair of opposite sides is congruent and parallel so given quadrilateral QUADis a parallelogram.

(c)

To determine

To state: The property that proves that the quadrilateral QUADwith its both pairs of opposite sides as equal, is a parallelogram.

Answer to Problem 1PSA

Both pairs of opposite sides in the given quadrilateral QUADare congruent to each other so it is a parallelogram.

Explanation of Solution

Given:In quadrilateral QUAD, $UQ\cong AD$ and $UA\cong QD$ .

Here both pairs of opposite sides are congruent to each other so the given quadrilateral QUADis a parallelogram.

(d)

To determine

To state: The property that proves that the quadrilateral QUADwith its both pairs of opposite sides as equal, is a parallelogram.

Answer to Problem 1PSA

In this case, none of the properties of a parallelogram is being satisfied by the given quadrilateral QUADso it is not a parallelogram.

Explanation of Solution

Given:In quadrilateral QUAD, $UQ\cong AD$ and $UA\parallel QD$ .

A quadrilateral can be a parallelogram only if it satisfies one of the given properties,

(i) If both pairs of opposite sides of a quadrilateral are parallel.

(ii) If both pairs of opposite sides of a quadrilateral are congruent.

(iii) If both pairs of opposite angles of a quadrilateral are congruent.

(iv)If the diagonals of a quadrilateral bisect each other.

(v) If one pair of opposite sides of a quadrilateral are both parallel and congruent.

But here none of the properties being satisfied so given quadrilateral QUADis not a parallelogram.

(e)

To determine

To state: The property that proves that the quadrilateral QUADwith a pair of opposite sides as parallel and a pair of alternate interior angles as equal, is a parallelogram.

Answer to Problem 1PSA

Both pairs of opposite sides in the given quadrilateral QUADare parallel to each other so it is a parallelogram.

Explanation of Solution

Given:In quadrilateral QUAD, $UQ\parallel AD$ and $\angle DQA\cong \angle UAQ$ .

As given,

  $\angle DQA\cong \angle UAQ$

  $\angle DQA$ and $\angle UAQ$ are alternate interior angles and if they are congruent then it implies that

  $UA\parallel QD$

Hence both pairs of opposite sides in the given quadrilateral QUADare parallel to each other so it is a parallelogram.