Chapter 6, Problem 4CUR

a.

To determine

To determine whether the given statement is always, sometimes, or never true for a parallelogram $ABCD$ with diagonals that intersect at $P$ .

Answer to Problem 4CUR

Sometimes true

Explanation of Solution

Given:

A parallelogram $ABCD$ with diagonals intersect at $P$ .

  

Statement: $AB\cong BC$

The given statement $AB\cong BC$ is sometimes true.

It will be true only when the parallelogram is a square.

Conclusion:

Therefore, the given statement is sometimes true.

b.

To determine

To determine whether the given statement is always, sometimes, or never true for a parallelogram $ABCD$ with diagonals that intersect at $P$ .

Answer to Problem 4CUR

Sometimes true

Explanation of Solution

Given:

A parallelogram $ABCD$ with diagonals intersect at $P$ .

  

Statement: $AC\perp BD$

The given statement $AC\perp BD$ is sometimes true only when the parallelogram is a rectangle.

Conclusion:

Therefore, the given statement is sometimes true.

c.

To determine

To determine whether the given statement is always, sometimes, or never true for a parallelogram $ABCD$ with diagonals that intersect at $P$ .

Answer to Problem 4CUR

Never true

Explanation of Solution

Given:

A parallelogram $ABCD$ with diagonals intersect at $P$ .

  

Statement: $\angle A$ and $\angle B$ are complementary angles.

The given statement $\angle A$ and $\angle B$ are complementary angles is never true because consecutive angles of a parallelogram are supplementary.

Conclusion:

Therefore, the given statement is never true.

d.

To determine

To determine whether the given statement is always, sometimes, or never true for a parallelogram $ABCD$ with diagonals that intersect at $P$ .

Answer to Problem 4CUR

Always true

Explanation of Solution

Given:

A parallelogram $ABCD$ with diagonals intersect at $P$ .

  

Statement: $\angle ADB\cong \angle CBD$

The given statement $\angle ADB\cong \angle CBD$ is always true.

Since $AB$ is parallel to $DC$ , so alternate interior angles are equal.

Conclusion:

Therefore, the given statement is always true.

e.

To determine

To determine whether the given statement is always, sometimes, or never true for a parallelogram $ABCD$ with diagonals that intersect at $P$ .

Answer to Problem 4CUR

Always true

Explanation of Solution

Given:

A parallelogram $ABCD$ with diagonals intersect at $P$ .

  

Statement: $AP\cong PC$

The given statement $AP\cong PC$ is always true as diagonals of a parallelogram bisect each other.

Conclusion:

Therefore, the given statement is always true.

f.

To determine

To determine whether the given statement is always, sometimes, or never true for a parallelogram $ABCD$ with diagonals that intersect at $P$ .

Answer to Problem 4CUR

Always true.

Explanation of Solution

Given:

A parallelogram $ABCD$ with diagonals intersect at $P$ .

  

Statement: $\Delta ABC\cong \Delta CDA$ .

The given statement $\Delta ABC\cong \Delta CDA$ is always true.

In $\Delta ABC$ and $\Delta CDA,$

  $AB=CD$ (given)

  $BC=DA$ (given)

  $AC=AC$ (common)

  $\therefore \Delta ABC\cong \Delta CDA$ by Side-Side-Side (SSS) congruency criteria.

Conclusion:

Therefore, the given statement is always true.