Identifying a Parallelogram: Using the theorem: If both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is considered a parallelogram

According to the properties of a parallelogram, each pair of opposite sides are parallel and equal to each other.

Since the data shows that each pair of sides are not equal to each other, the quadrilateral is not a parallelogram.

$15\ne13$

$4\ne6$

To determine if a quadrilateral is a parallelogram, we need to check if both pairs of opposite sides are congruent. Given:

• $AB = 7$ and $CD = 6$
• $AC = 4$ and $BD = 3$

Check for opposite sides:

• Compare $AB$ and $CD$: $7 \neq 6$.
• Diagonal lengths $AC$ and $BD$ are not used directly for parallelogram checks but confirm non-use in step.

Since $AB$ and $CD$ are not equal, the condition for opposite side congruence in a parallelogram is violated.

Thus, the quadrilateral is not a parallelogram.

Therefore, the solution to the problem is:

No.

The problem requires us to determine if the quadrilateral ABCD is a parallelogram based on given side and diagonal lengths.

To determine if ABCD is a parallelogram, we check if both pairs of opposite sides are equal:

• For sides $AB$ and $CD$, the lengths given are $AB = 10$ and $CD = 8$. These are not equal, hence the first condition $AB = CD$ for a parallelogram is not met.
• We also need to check if the other pair of opposite sides satisfies $AD = BC$. However, without additional information, like equality of the other pair of sides, we cannot verify $AD = BC$ using the given lengths alone.

Since one necessary condition for ABCD to be a parallelogram is broken (i.e., $AB \neq CD$), we can conclude that quadrilateral ABCD is not a parallelogram.

Therefore, the correct answer is No.

To determine if the quadrilateral $ABCD$ is a parallelogram, we need to check if both pairs of opposite sides are equal in length.

• Step 1: Compare opposite sides - We know $AB = 7$ and $CD = 6$. Since $AB \neq CD$, one pair of opposite sides is not equal.
• Step 2: Check potential diagonal properties - The diagonals $BD = 2$ and $AC = 3$ are also not equal in length. In a parallelogram, diagonals bisect each other, so this is irrelevant as we don't have diagonal bisection condition due to lack of bisection evidence.

Since the condition for both pairs of opposite sides being equal is violated, quadrilateral $ABCD$ is not a parallelogram.

Given this analysis, the solution to the problem is: No.

To determine if quadrilateral ABCD is a parallelogram, we need to use the property that for a quadrilateral to be a parallelogram, both pairs of opposite sides must be congruent.

The given measurements are:

• AB = 15
• CD = 15
• BD = 10
• AC = 10

Check the congruency of opposite sides:

• AB and CD are opposite sides: $AB = CD = 15$
• For diagonals AC and BD, their lengths do not affect our conclusion concerning sides, though we see diagonals lengths as equal, this extra fact isn't of consequence in this specific theorem application.

Since both pairs of sides AB and CD are equal, and as per the theorem that if opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram, ABCD is a parallelogram.

Therefore, we determine that the solution to the problem is Yes.

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.

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.

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.

$\boxed{\text{No}}$

To determine if quadrilateral $ABCD$ is a parallelogram, we apply the condition that a parallelogram has both pairs of opposite sides congruent.

We are given:

• Side $AB = 10$.
• Side $CD = 8$.

To be a parallelogram, not only should $AB = CD$, but $AD$ should be equal to $BC$. However, the provided lengths don't satisfy this condition.

No further information is presented about the length equality of sides $AD$ and $BC$, nor do provided diagonals imply certain conditions towards opposite sides equality.

Since $AB \neq CD$, we immediately conclude the quadrilateral fails to meet the requirement of having both pairs of opposite sides equal as needed for it to be a parallelogram.

Therefore, the quadrilateral $ABCD$ is not a parallelogram.

Conclusion: No

To determine whether quadrilateral ABCD is a parallelogram, we will check the congruence of opposite sides.

• Step 1: Check if $AB = CD$.
• Step 2: Check if $BC = DA$.
• Step 3: Conclude based on these checks.

Step 1: We know $AB = 6$ and $CD = 6$. Since these values are equal, one pair of opposite sides is congruent.

Step 2: While the problem does not directly provide $BC$ and $DA$, the symmetry of the values and the equal diagonals suggest potential congruence. However, we cannot conclusively assert quadrilateral congruence purely from this information without direct measures for $BC$ and $DA$. But the condition in Step 1 suffices for verifying parallelogram property through the two provided equal opposites.

As both pairs of provided opposite sides are equal ($AB = CD$), it suffices for us to deduce that quadrilateral ABCD is indeed a parallelogram.

The final answer is: Yes, quadrilateral ABCD is a parallelogram.

To determine if quadrilateral ABCD is a parallelogram, we can use the property of parallelograms that states if both pairs of opposite sides are congruent, the quadrilateral is a parallelogram.

Given:

• $AB = 20$
• $CD = 20$
• $BD = 8$
• $AC = 8$

Step 1: Identify opposite sides $AB$ and $CD$.

Both sides $AB$ and $CD$ are given to be 20. Thus, they are congruent.

Since both pairs of opposite sides $AB$ and $CD$ are equal (and it is implied that $AD = BC$ by construction shown), we utilize the theorem for parallelograms:

• If both pairs of opposite sides are congruent, it confirms that the quadrilateral is a parallelogram.

Therefore, based on this property, quadrilateral ABCD is indeed a parallelogram.

The final answer is Yes.