Parallelogram Verification: Quadrilateral with Sides 12 and Diagonals 6

Q: What's the difference between sides and diagonals in this problem?

Sides are the edges of the quadrilateral: AB, BC, CD, and DA. Diagonals are segments connecting opposite vertices: AC and BD. In this problem, AC = 6 and BD = 6 are diagonal lengths!

Q: Why do we only need to check opposite sides?

The parallelogram theorem states that if both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. We don't need to check all properties - just this one!

Q: How do I identify which sides are opposite?

In quadrilateral ABCD, opposite sides are: AB opposite to CD and BC opposite to AD. Think of it like a rectangle - top/bottom and left/right are opposite pairs.

Q: What if the diagonals weren't equal?

Diagonal equality doesn't matter for proving it's a parallelogram! Only opposite side equality is needed. Equal diagonals would make it a rectangle, but that's a different property.

Parallelogram Properties with Diagonal Measurements

Shown below is the quadrilateral ABCD.

AB = 12 and CD = 12.

BD = 6

AC = 6

Is the quadrilateral a parallelogram?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the square a parallelogram?

00:03 A pair of opposite sides are equal according to the given data

00:07 A second pair of opposite sides are equal according to the given data

00:12 A parallelogram is a quadrilateral with 2 pairs of equal opposite sides

00:15 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Shown below is the quadrilateral ABCD.

AB = 12 and CD = 12.

BD = 6

AC = 6

Is the quadrilateral a parallelogram?

Step-by-step solution

We need to determine if quadrilateral $ABCD$ is a parallelogram based on the side lengths and properties provided. For a quadrilateral to be a parallelogram, one way is to confirm whether both pairs of opposite sides are congruent.

We are given the following side lengths:

$AB = 12$
$CD = 12$
$AC = 6$
$BD = 6$

Let's apply the theorem: If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.

Checking the pairs of opposite sides:

Side $AB = 12$ and side $CD = 12$ . Thus, $AB = CD$ .
Side $AC = 6$ and side $BD = 6$ . Thus, $AC = BD$ .

Since both pairs of opposite sides are congruent, quadrilateral $ABCD$ satisfies the parallelogram condition.

Therefore, the answer to the question is that the quadrilateral is indeed a parallelogram, and the correct choice is:

Yes.

Final Answer

Yes.

Key Points to Remember

Essential concepts to master this topic

Rule: Parallelogram has both pairs of opposite sides congruent
Technique: Check AB = CD and BC = AD: 12 = 12 ✓
Check: Verify opposite sides are equal: AB = CD = 12, BC = AD = 6 ✓

Common Mistakes

Avoid these frequent errors

Confusing diagonals with sides
Don't assume AC and BD are sides = wrong identification! These are diagonal lengths, not side lengths. Always identify which segments are sides (AB, BC, CD, DA) versus diagonals (AC, BD).

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's the difference between sides and diagonals in this problem?

Sides are the edges of the quadrilateral: AB, BC, CD, and DA. Diagonals are segments connecting opposite vertices: AC and BD. In this problem, AC = 6 and BD = 6 are diagonal lengths!

Why do we only need to check opposite sides?

The parallelogram theorem states that if both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram. We don't need to check all properties - just this one!

How do I identify which sides are opposite?

In quadrilateral ABCD, opposite sides are: AB opposite to CD and BC opposite to AD. Think of it like a rectangle - top/bottom and left/right are opposite pairs.

What if the diagonals weren't equal?

Diagonal equality doesn't matter for proving it's a parallelogram! Only opposite side equality is needed. Equal diagonals would make it a rectangle, but that's a different property.

Can I use other methods to prove it's a parallelogram?

Yes! You could also prove it by showing:

One pair of opposite sides are both parallel and congruent
Both pairs of opposite sides are parallel
Diagonals bisect each other

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Parallelogram Verification: Quadrilateral with Sides 12 and Diagonals 6

Parallelogram Properties with Diagonal Measurements

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What's the difference between sides and diagonals in this problem?

Why do we only need to check opposite sides?

How do I identify which sides are opposite?

What if the diagonals weren't equal?

Can I use other methods to prove it's a parallelogram?

🌟 Unlock Your Math Potential

More Questions

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Quadrilateral Analysis: Testing Parallelogram Properties with Side Lengths 7, 6, 2, and 3