Parallelogram Verification: Comparing Sides x+5 and 2x+9

Q: Why can't I just use the fact that AB = DC = 8?

A parallelogram requires both pairs of opposite sides to be equal. Having just one pair equal (AB = DC) isn't enough - you also need AD = BC to confirm it's a parallelogram.

Q: How do I find the value of x to substitute?

Look at the diagram! You can see that AB = x = 8 and DC = 8, so x must equal 8. Use this value to calculate the other sides.

Q: What if AD and BC were equal - would it be a parallelogram then?

Yes! If both pairs of opposite sides are equal (AB = DC AND AD = BC), then the quadrilateral would definitely be a parallelogram.

Q: Why do I substitute x = 8 into both AD and BC?

The variable x appears in both expressions: AD = 2x + 9 and BC = x + 5. Since x = 8 (from AB = x), substitute this same value into both to find their actual lengths.

Parallelogram Properties with Variable Side Lengths

AB = DC.=

Is the shape below a parallelogram?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

AB = DC.=

Is the shape below a parallelogram?

Step-by-step solution

In a parallelogram, we know that each pair of opposite sides are equal to each other.

The data shows that only one pair of sides are equal to each other:

$AB=DC=8$

Now we try to see that the additional pair of sides are equal to each other.

We replace $x=8$ for each of the sides:

$AD=2\times8+9$

$AD=16+9$

$AD=25$

$BC=8+5$

$BC=13$

That is, we find that the pair of opposite sides are not equal to each other:

$25\ne13$

Therefore, the quadrilateral is not a parallelogram.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Parallelogram Rule: Both pairs of opposite sides must be equal
Variable Substitution: Use x = 8 to find AD = 25 and BC = 13
Verification Check: Compare opposite sides: if 25 ≠ 13, then not a parallelogram ✓

Common Mistakes

Avoid these frequent errors

Checking only one pair of opposite sides
Don't assume the quadrilateral is a parallelogram if only AB = DC = 8! This only shows one pair is equal. Always check that BOTH pairs of opposite sides are equal: AB = DC AND AD = BC.

Practice Quiz

Test your knowledge with interactive questions

It is possible to draw a quadrilateral that is not a rectangle and that has two equal opposite sides?

FAQ

Everything you need to know about this question

Why can't I just use the fact that AB = DC = 8?

A parallelogram requires both pairs of opposite sides to be equal. Having just one pair equal (AB = DC) isn't enough - you also need AD = BC to confirm it's a parallelogram.

How do I find the value of x to substitute?

Look at the diagram! You can see that AB = x = 8 and DC = 8, so x must equal 8. Use this value to calculate the other sides.

What if AD and BC were equal - would it be a parallelogram then?

Yes! If both pairs of opposite sides are equal (AB = DC AND AD = BC), then the quadrilateral would definitely be a parallelogram.

Can a quadrilateral be a parallelogram with just parallel sides?

Parallel sides alone aren't enough from the given information. We use the opposite sides equal test here because we have specific side lengths, not angle measures.

Why do I substitute x = 8 into both AD and BC?

The variable x appears in both expressions: AD = 2x + 9 and BC = x + 5. Since x = 8 (from AB = x), substitute this same value into both to find their actual lengths.

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