Parallelogram Verification: Quadrilateral ABCD with Sides 15, 12, 7, and 8

Q: What makes a quadrilateral a parallelogram?

A quadrilateral is a parallelogram when both pairs of opposite sides are equal. For ABCD, you need AB = CD and BC = AD. If even one pair isn't equal, it's not a parallelogram.

Q: Are AC and BD opposite sides?

No! AC and BD are diagonals (they connect opposite vertices). The opposite sides are AB & CD, and BC & AD. Don't confuse diagonals with sides when checking parallelogram properties.

Q: Do I need all four side lengths to verify?

You need to know the lengths of opposite sides to compare them. In this problem, we have AB=15, CD=12, AC=8, BD=7. Since AC and BD are diagonals, we can't fully verify without knowing BC and AD lengths.

Q: What if only one pair of opposite sides is equal?

That's not enough for a parallelogram! You need both pairs of opposite sides to be equal. Having just one pair equal might make it a trapezoid, but not a parallelogram.

Parallelogram Verification with Side Length Comparison

Look at the quadrilateral ABCD.

AB = 15

CD = 12

BD = 7

AC = 8

Is this quadrilateral a parallelogram?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Our question is: Is the quadrilateral a parallelogram?

00:18 A parallelogram has two pairs of equal opposite sides.

00:22 Let's check the sides of our quadrilateral now.

00:26 Oh! The opposite sides aren't equal, so it's not a parallelogram.

00:30 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the quadrilateral ABCD.

AB = 15

CD = 12

BD = 7

AC = 8

Is this quadrilateral a parallelogram?

Step-by-step solution

To determine whether the quadrilateral ABCD is a parallelogram, we must establish whether both pairs of opposite sides are equal. In a parallelogram, this criterion is met: the opposite sides are congruent.

First, we compare the lengths of the opposite sides:

Length $AB = 15$ and length $CD = 12$ . These two sides are not equal.
Length $AC = 8$ and length $BD = 7$ . These two sides are also not equal.

Since neither pair of opposite sides is equal, quadrilateral ABCD does not satisfy the conditions for being a parallelogram.

Therefore, the solution to the problem is that quadrilateral ABCD is not a parallelogram.

$\boxed{\text{No}}$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Parallelograms have both pairs of opposite sides equal
Technique: Compare AB=15 vs CD=12 and AC=8 vs BD=7
Check: If any pair of opposite sides unequal, not a parallelogram ✓

Common Mistakes

Avoid these frequent errors

Confusing diagonals with sides
Don't compare diagonal lengths AC and BD as if they were opposite sides = wrong conclusion! These are diagonals, not sides of the quadrilateral. Always compare actual opposite sides: AB with CD, and BC with AD.

Practice Quiz

Test your knowledge with interactive questions

Shown below is the quadrilateral ABCD.

AB = 15 and CD = 13.

BD = 6 and AC = 4

Is it possible to conclude that this quadrilateral is a parallelogram?

FAQ

Everything you need to know about this question

What makes a quadrilateral a parallelogram?

A quadrilateral is a parallelogram when both pairs of opposite sides are equal. For ABCD, you need AB = CD and BC = AD. If even one pair isn't equal, it's not a parallelogram.

Are AC and BD opposite sides?

No! AC and BD are diagonals (they connect opposite vertices). The opposite sides are AB & CD, and BC & AD. Don't confuse diagonals with sides when checking parallelogram properties.

Do I need all four side lengths to verify?

You need to know the lengths of opposite sides to compare them. In this problem, we have AB=15, CD=12, AC=8, BD=7. Since AC and BD are diagonals, we can't fully verify without knowing BC and AD lengths.

What if only one pair of opposite sides is equal?

That's not enough for a parallelogram! You need both pairs of opposite sides to be equal. Having just one pair equal might make it a trapezoid, but not a parallelogram.

Can I use the diagonal lengths to check anything?

Diagonal lengths alone don't determine if it's a parallelogram. However, in a parallelogram, diagonals bisect each other (cut each other in half). But this requires knowing where they intersect, not just their lengths.

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