Chapter 6.1, Problem 3WE

a.

To determine

To determine whether the given statement can be deduced from what is given.

Answer to Problem 3WE

No

Explanation of Solution

Given:

A parallelogram $ABCD$

  $AC>BD$

  

Given statement: $AB>AD$

Calculation:

The opposite sides of a parallelogram are congruent and parallel.

Since $ABCD$ is a parallelogram, no information is given regarding the sides of the parallelogram.

So the statement $AB>AD$ cannot be deduced from the given information.

Conclusion:

No, the statement $AB>AD$ cannot be deduced from the given statement.

b.

To determine

To determine whether the given statement can be deduced from what is given.

Answer to Problem 3WE

No

Explanation of Solution

Given:

A parallelogram $ABCD$

  $AC>BD$

  

Given statement: $AM>MC$

The diagonals of a parallelogram bisect each other.

Since $ABCD$ is a parallelogram, its diagonals bisect each other.

So, $AM=MC$ is possible but $AM>MC$ is not possible.

Conclusion:

No, the given statement $AM>MC$ cannot be deduced from the given information.

c.

To determine

To determine whether the given statement can be deduced from what is given.

Answer to Problem 3WE

Yes

Explanation of Solution

Given:

A parallelogram $ABCD$

  $AC>BD$

  

Given statement: $DM=MB$

Calculation:

The diagonals of a parallelogram bisect each other.

Since $ABCD$ is a parallelogram, its diagonals bisect each other.

So, $DM=MB$ .

Conclusion:

Yes, the given statement $DM=MB$ can be deduced from the given information.

b.

To determine

To determine whether the given statement can be deduced from what is given.

Answer to Problem 3WE

Yes

Explanation of Solution

Given:

A parallelogram $ABCD$

  $AC>BD$

  

Given statement: $AM>MB$

Calculation:

The diagonals of a parallelogram bisect each other.

Since, $ABCD$ is a parallelogram and its diagonals bisect each other.

So, $AM=MC=\frac{1}{2}AC$ ….(i).

And $DM=MB=\frac{1}{2}DB$ ….(ii).

So, $AC>BD$ (given),

  $\frac{1}{2}AC>\frac{1}{2}BD$ (multiplying by $\frac{1}{2}$ on both sides),

From (i) and (ii)

  $\Rightarrow AM>MB$

Conclusion:

Yes, the given statement $AM>MB$ can be deduced from the given information.