Properties, Characteristics and proofs: Determine whether the quadrilateral is a square

The problem requires checking whether quadrilateral $ABCD$ is a square.

• The quadrilateral is labeled such that each side $AB$, $BC$, $CD$, and $DA$ is $6$, indicating that all sides are of equal length.
• In geometry, if a quadrilateral has all sides equal and all internal angles as right angles, it is defined as a square.
• Considering all sides are given as equal, it's logical to presume the angles in this labeled quadrilateral are right angles, a common property when the shape is indeed a square.
• With both conditions—equal sides and right angles—met, we conclude the figure is a square.

Therefore, the quadrilateral $ABCD$ is a square.

Yes

As we see that BD is equal to 8 and AC is equal to 7, the sides are not equal, and this contradicts the properties of the square, where all sides are equal to each other, therefore the quadrilateral is not a square

To determine if quadrilateral ABCD is a square, we need to verify if all sides are equal and if all angles are $90^\circ$.

1. Given side lengths are $AB = 3$, $BC = 6$, $CD = 3$, and $DA = 6$. Since all sides are not equal, ABCD cannot be a square.

2. Even without needing to calculate angles, we know that a square requires all four sides to be equal, which they are not. Therefore, this characteristic alone is sufficient to conclude.

Since we determined the side lengths don't match, it logically follows that ABCD is not a square.

Therefore, the solution to the problem is No.

To determine if the quadrilateral ABCD is a square, we must check the properties of a square. A square requires all four sides to have equal lengths and all interior angles to be $90^\circ$.

• Step 1: Identify the given side lengths.

• Step 2: Compare the side lengths.

• Step 3: Evaluate the conclusion based on side length comparison.

Step 1: The diagram provides us with the following side lengths:

• AB = 8

• BC = 5

Step 2: Compare the given sides.

The side AB is labeled with a length of 8, while the vertical side BC is labeled 5. For ABCD to be a square, all sides would need to be the same length.

Step 3: Evaluation.

Since two adjacent sides AB and BC have different lengths (AB = 8 and BC = 5), it is evident that not all sides are equal.

As a result, we can conclude that ABCD cannot be a square because the sides are not all the same length.

Therefore, the quadrilateral ABCD is not a square.

To determine if the quadrilateral $ABCD$ is a square, we need to verify two main criteria:

• All sides must be equal: The sides provided are $AB = BC = CD = DA = 5$. This condition is satisfied.
• All angles must be $90^\circ$: There is no clear indication or evidence presented in the problem that the angles are right angles.

Since we have confirmed that all sides are equal but lack the confirmation for all angles being $90^\circ$, we cannot assume $ABCD$ is a square. In a typical analysis without explicit angle data or additional geometric hints, equality of angles should not be assumed solely based on the visual diagram or equal sides.

Therefore, without concrete proof of all angles being $90^\circ$, the quadrilateral $ABCD$ is not definitively a square.

Therefore, the answer to the problem is "No".

To determine if the quadrilateral ABCD is a square, we must confirm two simple properties: all sides are equal, and all angles are right angles.

According to the problem, ABCD is a quadrilateral where all sides denoted by $\overline{AB} = \overline{BC} = \overline{CD} = \overline{DA} = 3$. This indicates that all sides are equal.

Visual context and information suggest, despite angles not being numerically specified, that the shape illustrates a square. Typically, when all sides are equal and we assume connecting lines that meet perpendicularly (which a standard square depiction strongly suggests), the angles are $90^{\circ}$.

Therefore, since all sides are equal and it is reasonable to expect all angles are right angles from the context provided, ABCD satisfies the conditions for being a square.

Thus, the quadrilateral ABCD is a square, so the correct answer is "Yes".

To determine whether quadrilateral $ABCD$ is a square, we need to check two main properties:

• All four sides must be of equal length.
• All angles must be $90^\circ$.

In the context of the given geometrical figure, even though the diagram appears as a quadrilateral within a square shape, the internal details such as equal side lengths and angle measures are not specified or labeled.

Without confirmation of equal side lengths and precise angle measurements from the problem, we cannot definitively classify $ABCD$ as a square based solely on appearance. For a definitive determination, specific geometric calculations or measurements are required.

Therefore, $ABCD$ is not a square, without concrete evidence supporting the properties of a square.

The correct answer to the problem is no.

To determine if the quadrilateral is a square, we need to verify the defining characteristics of a square:

• All four sides must be of equal length.
• Each of the four angles must be a right angle ($90^\circ$).
• The diagonals should be equal and bisect each other at right angles.

From the problem description and the diagram provided:

• The quadrilateral is labeled ABCD, and the diagram shows right angles at each corner ($\angle A = \angle B = \angle C = \angle D = 90^\circ$).
• Visual inspection of the diagram suggests the sides AB, BC, CD, and DA appear equal, conforming to the property that all sides of a square are equal. This is inferred based on the symmetry and right angles in the diagram.
• Furthermore, the diagonals AC and BD in the diagram intersect at right angles, suggesting they are equal and bisect each other, which is a characteristic property of a square.

Therefore, checking all the conditions:

• Equal sides — based on the diagram.
• Right angles at each vertex — confirmed as all angles are labeled $90^\circ$.
• Equal and bisected diagonals — inferred from the intersection and right angles.

Conclusively, the quadrilateral satisfies all the characteristics of a square. Thus, the given quadrilateral is indeed a square.

Yes, the shape is a square.