Properties, Characteristics and proofs: True / false

To solve this problem, we need to understand the definitions and properties of a parallelogram and a square:

• A parallelogram is a quadrilateral where opposite sides are parallel. This implies that opposite sides are equal in length, but it does not require all sides to be equal or all angles to be right angles.
• A square is a special type of parallelogram and rectangle where all four sides are of equal length, and all four angles are right angles (90 degrees).

With these definitions in mind, let's compare:

A parallelogram, by definition, does not require all sides to be equal or all angles to be right angles. Therefore, not every parallelogram meets the requirements to be a square.

For example, a rectangle is a type of parallelogram where all angles are right angles, but it may not have all equal sides unless it is a square. Similarly, a rhombus is a type of parallelogram with all sides equal but may not have all right angles unless it is a square.

Thus, while a square is indeed a parallelogram (since it fulfills the conditions of having opposite sides equal and parallel), not every parallelogram is a square. Only those parallelograms which have all sides equal and all angles equal to 90 degrees qualify as squares.

This leads us to conclude that the statement "A parallelogram is a square" is false.

Therefore, the correct answer is No.

To determine if a square is a parallelogram, we must first define both geometric shapes.

• Square: A square is a quadrilateral with four equal sides and four right angles. This means that all angles are $90^\circ$ and each pair of opposite sides are parallel.
• Parallelogram: A parallelogram is a quadrilateral where opposite sides are parallel and equal in length. It does not necessarily require right angles.

Now, let's see if a square fits the definition of a parallelogram:

• The square has opposite sides that are both parallel and equal, satisfying the definition of a parallelogram.
• Although a square also has additional properties, such as all angles being right angles and all sides being equal, these characteristics do not contradict the definition of a parallelogram.

Since a square satisfies all the conditions required for a parallelogram, we conclude that a square is indeed a type of parallelogram.

Therefore, the answer to the problem is Yes.

To determine whether a rhombus is a square, we must understand the properties of each shape.

Definition of a Rhombus:
A rhombus is a quadrilateral with all four sides of equal length. It may have angles that are not right angles.

Definition of a Square:
A square is a quadrilateral with all four sides of equal length and all four angles equal to $90^{\circ}$.

Comparison:

• Both a square and a rhombus have four equal sides.
• A square has $90^{\circ}$ angles, whereas a rhombus does not necessarily have $90^{\circ}$ angles.

Therefore, a rhombus is not a square as a general statement.

To solve this problem, we'll consider the definitions:

• A square is a quadrilateral with four equal sides and four equal angles (each angle is $90$ degrees).
• A rhombus is a quadrilateral with all sides equal in length.

Notice that for a quadrilateral to be a rhombus, it simply requires all sides to be equal, without any condition on the angles. Since a square has all four sides equal, it meets the fundamental requirement of a rhombus.

Therefore, every square can be classified as a rhombus because it satisfies the condition that all sides are equal.

Hence, the correct answer is Yes, the square is a rhombus.

In this problem, we need to determine if a square meets the criteria for being classified as a rectangle. We start by examining the definitions:

• Definition of a Rectangle: A rectangle is a quadrilateral with four right angles. Additionally, opposite sides are equal in length.
• Definition of a Square: A square is a quadrilateral where all four sides are equal in length, and all four angles are right angles.

By examining these properties, we can see the following:

• Since a square has four right angles, it satisfies the angle condition of a rectangle.
• A square has all sides equal, which means opposite sides are also equal. This satisfies the side condition of a rectangle.

Therefore, since a square fulfills both the angle and opposite sides conditions required by the definition of a rectangle, a square is indeed a rectangle.

The correct answer to the question is: Yes.

To solve this problem, we'll identify key properties of a square and a trapezoid:

• Step 1: Define a trapezoid. A trapezoid is a quadrilateral with at least one pair of parallel sides.
• Step 2: Define a square. A square is a quadrilateral with four equal sides and four right angles, characterized by having two pairs of parallel sides.
• Step 3: Compare properties. A square inherently meets the trapezoid's criterion, because it has two pairs of parallel sides. However, the converse is not true—a trapezoid does not necessarily have four equal sides and four right angles.

Now, let's elaborate:
Step 1: A trapezoid (or trapezium in some regions) is defined primarily by having only one pair of parallel sides. This means a trapezoid does not require all sides to be equal or to have right angles.
Step 2: A square, on the other hand, has stricter requirements: all sides must be equal in length and each angle must be a right angle (90 degrees). This ensures that the square also qualifies as a rhombus and a rectangle, given its properties.
Step 3: When we compare the two, while every square can be technically considered a trapezoid (since it fulfills the base condition of having parallel sides), not every trapezoid can be seen as a square because it lacks the requirement for equal sides and right angles.
Therefore, the question of whether a trapezoid is a square can be answered simply by verifying these fundamental geometric characteristics.

With these points in mind, the correct answer is:

No, a trapezoid cannot be classified as a square.

To determine if a square is a trapezoid, we need to understand the definitions of both shapes:

• Definition of a Trapezoid: A trapezoid is a type of quadrilateral that has at least one pair of parallel sides.
• Definition of a Square: A square is a quadrilateral with four equal sides and four right angles, and it has two pairs of parallel sides (opposite sides are parallel).

Since a square has two pairs of parallel sides, it certainly has at least one pair of parallel sides, which satisfies the definition of a trapezoid under the inclusive definition.

Therefore, we conclude that a square is indeed a trapezoid.

The correct answer to the question is: Yes.

To determine if a deltoid is a square, we need to examine the defining properties of both shapes:

• Square: A square is a quadrilateral with four equal sides and four equal angles of $90^\circ$.
• Deltoid (Kite): A deltoid is a quadrilateral with two distinct pairs of adjacent sides that are equal. It does not necessarily have right angles and does not require opposite sides to be equal.

Upon comparing these definitions, we can see the differences:

• A square requires all sides to be equal and all angles to be $90^\circ$.
• A deltoid, while having two pairs of adjacent equal sides, does not require all sides to be equal, nor does it need to have $90^\circ$ angles.

Given these properties, it is clear that while all squares can be seen as a special type of deltoid (specifically when two adjacent pairs of sides are equal), not all deltoids are squares because they lack the requirement for right angles and equal opposite sides.

Therefore, the answer to the question "Is a deltoid a square?" is No.

To determine if a square is also a deltoid, let's analyze the properties of both shapes:

• A square is a quadrilateral with all four sides equal and all angles right angles ($90^\circ$).
• A deltoid (or kite) is defined as a quadrilateral with two distinct pairs of adjacent sides that are equal.

Now, consider a square:

• In a square, all four sides are equal: this means it has two pairs of adjacent sides that are equal because any pair of adjacent sides are equal (all sides are equal).

Since a square indeed has these two pairs of adjacent equal sides, it satisfies the definition of a deltoid. Therefore, in the context of these definitions, a square can indeed be classified as a deltoid.

Therefore, the correct answer to whether a square is a deltoid is Yes.

Let's think about the different types of lines.

Looking at side AB and side CD, we can see that if we extend both of them, they will never intersect.

Also, according to the properties of a rectangle, each pair of opposite sides are parallel to each other.

Therefore, the answer is correct and indeed AB is parallel to CD.

We know that the figure shows a square and that, in a square, every pair of opposite sides are parallel.

We also know that every pair of opposite sides in a rectangle are parallel as well.

Therefore, the quadrilateral ABCD is indeed a rectangle.

Firstly, let's think about the properties of a square—remembering that all sides are equal to each other.

In other words, all sides in the square are equal to 4 cm. As a result, we can determine that the distance between BD and AC is indeed equal to 4 since CD is also equal to 4.

In a square all sides are equal to one another, meaning that all the sides measure 5 cm.

Furthermore in a square, all angles are equal, they are right angles and measure 90 degrees.

Therefore, we can determine that the diagonal divides each angle into 2 angles of 45 degrees, thus the diagonals must intersect.