Characteristics and Proofs of a Rectangle: True / false

We know that the sum of the angles in a quadrilateral is 360.

Since a rectangle is a type of quadrilateral, the sum of its angles is also equal to 360.

Likewise, we know that all the angles in a rectangle are right angles (90 degrees).

$90\times4=360$

To determine whether the given statement is true or false, we must rely on the fundamental properties of a rectangle.

A rectangle is a type of quadrilateral that has the following defining properties:

• All four interior angles are right angles, each measuring $90^\circ$.
• The opposite sides are parallel and equal in length.

Based on these properties, each angle in a rectangle is always a right angle. Therefore, contrary to the statement given, a rectangle does not have only one right angle; in fact, it has four right angles.

Thus, the statement "There is only one right angle in a rectangle" is False.

To solve this problem, we'll follow these steps:

• Identify the diagonal lines of the rectangle $AC$ and $BD$.
• Apply the diagonal property theorem for rectangles.

Let's go through the solution:

A rectangle in Euclidean geometry has several properties, one of which is that its diagonals are equal in length. This can be derived from the fact that opposite sides of a rectangle are equal and parallel, making it a specific type of parallelogram. Thus, both diagonals split the rectangle into congruent triangles, ensuring they are of equal length.

For rectangle $ABCD$, diagonals $AC$ and $BD$ must be equal due to this inherent geometric property.

Therefore, the statement that "the diagonals of rectangle ABCD are equal" is True.

To determine whether the diagonals of rectangle $ABCD$ are perpendicular, we need to recall the geometric properties of a rectangle. In a rectangle, the diagonals are congruent, which means they have the same length, but they are not inherently perpendicular. An exception occurs only if the rectangle is also a square, as squares have perpendicular diagonals.

Let's review the properties:

• Each diagonal divides the rectangle into two congruent right triangles.
• The diagonals of a rectangle are equal in length: $AC = BD$.
• The diagonals are not perpendicular; they do not intersect at right angles unless the rectangle is a square.

Since there is no information provided that suggests rectangle $ABCD$ is a square, we conclude based on standard rectangle properties that the diagonals $AC$ and $BD$ are not perpendicular.

Thus, the statement "The diagonals of rectangle $ABCD$ are perpendicular to each other" is False.

In any typical rectangle, the diagonals intersect and bisect each other, meaning they divide each other into two equal parts. However, they do not bisect the angles of the rectangle. This is a characteristic property of the diagonals in a rhombus or square, where each diagonal does indeed bisect the angles from which it extends.

To further understand this, let's analyze the diagonal behavior:

• The diagonals of a rectangle are equal in length and split the rectangle into two congruent right-angled triangles.
• These triangles have angles which consist of the original rectangle's angles and two parts from the diagonal intersect. However, the diagonals do not split the original angles equally.
• If the diagonals of a rectangle properly bisected the rectangle's angles, it would mean each angle at the vertices (e.g., the angles at A, B, C, D) is divided equally into two smaller angles. But this is not true for rectangles, as the diagonals merely act as chords intersecting the rectangle, forming unequal angles unless it is a square.

Therefore, it's crucial to recognize that while a rectangle's diagonals bisect each other, they do not bisect the rectangle's angles unless the rectangle is a square.

As such, the statement that "The diagonals of rectangle ABCD bisect its angles" is False.