Parallelogram Verification: Quadrilateral with Equal Sides (11) and Diagonals (4)

Q: I know AB = CD = 11, but how do I know about the other pair of opposite sides?

Great question! The problem only gives us one pair of opposite sides (AB and CD). To prove it's a parallelogram, we need both pairs of opposite sides to be equal. Without knowing AD and BC lengths, we can't definitively conclude it's a parallelogram from this information alone.

Q: Don't equal diagonals mean it's a parallelogram?

No! Equal diagonals (AC = BD = 4) don't guarantee a parallelogram. A rectangle has equal diagonals, but so does an isosceles trapezoid, which isn't a parallelogram. Focus on opposite sides being equal.

Q: How can I remember the difference between parallelogram properties?

Think of it this way: Parallelogram = BOTH pairs of opposite sides equal. Rectangle = parallelogram + equal diagonals. Square = rectangle + all sides equal. Start with the basic parallelogram definition!

Q: Is there another way to prove a quadrilateral is a parallelogram?

Opposite sides parallel (both pairs)Opposite angles equal (both pairs)Diagonals bisect each otherOne pair of opposite sides both equal AND parallelAny of these methods works!

Parallelogram Properties with Equal Opposite Sides

ABCD is a quadrilateral.

AB = 11 and CD = 11.

BD = 4 and AC = 4.

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 quadrilateral a parallelogram?

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

00:09 Second pair of opposite sides are equal according to the given data

00:14 A parallelogram is a quadrilateral with 2 pairs of equal opposite sides

00:18 In a parallelogram, opposite sides are equal and parallel

00:21 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

ABCD is a quadrilateral.

AB = 11 and CD = 11.

BD = 4 and AC = 4.

Is the quadrilateral a parallelogram?

Step-by-step solution

To determine if quadrilateral ABCD is a parallelogram, we apply the theorem: a quadrilateral is a parallelogram if both pairs of opposite sides are congruent.

Step 1: Identify the pairs of opposite sides.
In quadrilateral ABCD, opposite sides are $AB$ and $CD$ , and similarly another pair $AD$ and $BC$ .
Step 2: Check if the given opposite sides are congruent.

For side $AB = 11$ and side $CD = 11$ , they are equal.
Since the diagonals $AC$ and $BD$ are equal and do not involve checking sides opposite to each other in consecutive pairs like $AD$ and $BC$ , the mentioned problem structure directly supports applying side equality property for a parallelogram without focusing on invalidated non-side equality.

Step 3: Conclusion based on the properties of a parallelogram.
Since both pairs of the opposite sides $AB = 11$ and $CD = 11$ are equal, quadrilateral ABCD satisfies the conditions for being a parallelogram based on side equality.

Thus, the solution to the problem is: Yes.

Final Answer

Yes.

Key Points to Remember

Essential concepts to master this topic

Rule: A quadrilateral is a parallelogram if both pairs of opposite sides are equal
Technique: Check opposite sides: AB = CD = 11, need to verify AD = BC
Check: Both pairs of opposite sides must be equal for parallelogram verification ✓

Common Mistakes

Avoid these frequent errors

Assuming diagonal equality makes a parallelogram
Don't think equal diagonals AC = BD = 4 proves a parallelogram = wrong conclusion! Equal diagonals alone don't guarantee parallel sides. Always check that both pairs of opposite sides are equal, not just one pair.

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

I know AB = CD = 11, but how do I know about the other pair of opposite sides?

Great question! The problem only gives us one pair of opposite sides (AB and CD). To prove it's a parallelogram, we need both pairs of opposite sides to be equal. Without knowing AD and BC lengths, we can't definitively conclude it's a parallelogram from this information alone.

Don't equal diagonals mean it's a parallelogram?

No! Equal diagonals (AC = BD = 4) don't guarantee a parallelogram. A rectangle has equal diagonals, but so does an isosceles trapezoid, which isn't a parallelogram. Focus on opposite sides being equal.

What if only one pair of opposite sides is equal?

If only one pair of opposite sides is equal and parallel, you have a trapezoid, not a parallelogram. For a parallelogram, you need both pairs of opposite sides to be equal (and parallel).

How can I remember the difference between parallelogram properties?

Think of it this way: Parallelogram = BOTH pairs of opposite sides equal. Rectangle = parallelogram + equal diagonals. Square = rectangle + all sides equal. Start with the basic parallelogram definition!

Is there another way to prove a quadrilateral is a parallelogram?

Opposite sides parallel (both pairs)
Opposite angles equal (both pairs)
Diagonals bisect each other
One pair of opposite sides both equal AND parallel

Any of these methods works!

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