Question

dont forget!
●the definition of a square
a quadrangle where all four angles and four sides are equal is called a square
●the property of a square
the diagonals of a square are equal in length and intersect perpendicularly.
1 the following sentences describe a square.
fill the appropriate words in the blanks.
a square has both the property of a rectangle in which all four are equal and the property of
a rhombus in which all four are equal.
from the above, the square is a special form of, and parallelogram.
the diagonals of a square are equal in and intersect
2 answer the following questions about the four quadrangles
on the right.
(1) which quadrilateral has all four sides equal
to each other?
(ans.)
(2) which quadrilateral has diagonals that are
equal in length?
(ans.)
(3) which quadrilateral has diagonals that
intersect perpendicularly?
(ans.)
(4) quadrilateral d has the properties of both quadrilateral b and
quadrilateral c. list the properties of its sides and angles.
(ans.)

Explanation:

Problem 1 (Filling the blanks)

Step 1: Analyze the properties of a square related to rectangle

A square has the property of a rectangle where all four angles are equal (since a rectangle has four right angles, and a square is a special rectangle with equal sides too).

Step 2: Analyze the properties of a square related to rhombus

A square has the property of a rhombus where all four sides are equal (a rhombus has four equal sides, and a square is a special rhombus with right angles).

Step 3: Identify the special forms of a square

From the above, the square is a special form of rectangle, rhombus, and parallelogram (since both rectangles and rhombuses are parallelograms, and a square is a special case of both).

Step 4: Analyze the diagonals of a square

The diagonals of a square are equal in length and intersect perpendicularly (as given in the "Don't forget!" section about the property of a square).

• A rhombus (C) has all four sides equal. A square (D) is a special rhombus with right angles, so it also has four equal sides. A parallelogram (A) does not necessarily have all four sides equal, and a rectangle (B) has opposite sides equal but not all four sides equal in general (unless it's a square). So the quadrilaterals with all four sides equal are C (rhombus) and D (square).

• A rectangle (B) has diagonals of equal length (a property of rectangles). A square (D) is a special rectangle, so its diagonals are also equal in length. A parallelogram (A) and a rhombus (C) do not necessarily have equal - length diagonals (unless the rhombus is a square). So the quadrilaterals with diagonals equal in length are B (rectangle) and D (square).

Answer:

1. A square has both the property of a rectangle in which all four \(\boldsymbol{\text{angles}}\) are equal and the property of a rhombus in which all four \(\boldsymbol{\text{sides}}\) are equal.

From the above, the square is a special form of \(\boldsymbol{\text{rectangle}}\), \(\boldsymbol{\text{rhombus}}\), and parallelogram.
The diagonals of a square are equal in \(\boldsymbol{\text{length}}\) and intersect \(\boldsymbol{\text{perpendicularly}}\).

Problem 2 (Answering questions about quadrilaterals)

(1) Which quadrilateral has all four sides equal to each other?