Question

3 in the following figures (1), (2), (3), and (4) only the diagonals of a certain quadrilateral are shown. choose either parallelogram, rectangle, rhombus, or square for the diagonals of each quadrilateral below. be sure to use each shape only once.
4 in the explanation of the following quadrilaterals, write ○ for the correct description, and × for the incorrect description of each shape.
(1) squares are rhombuses. ……………………………………………………………………………………………
(2) rhombuses are rectangles. ………………………………………………………………………………………
(3) rectangles are squares. ……………………………………………………………………………………………
(4) both rectangles and rhombuses are parallelograms. ……………………………………………………
(5) squares are rectangles, rhombuses, and parallelograms. ……………………………………………
(6) quadrilaterals that are neither rectangles nor rhombuses are not parallelograms. ……………

Answer:

Question 3 (Classifying Quadrilaterals by Diagonals)

Step 1: Recall Properties of Diagonals

• Parallelogram: Diagonals bisect each other (no other restrictions).
• Rectangle: Diagonals are equal in length and bisect each other.
• Rhombus: Diagonals are perpendicular and bisect each other (sides are equal).
• Square: Diagonals are equal, perpendicular, and bisect each other (all sides equal).

Figure (1)

Diagonals bisect each other (2 cm and 6 cm segments), but are not equal or perpendicular. This matches a parallelogram (since only bisecting is required).

Figure (2)

Diagonals bisect each other (4 cm and 5 cm segments) and are equal in length? Wait, no—wait, the segments are 4 cm and 5 cm? Wait, no, the diagram shows 4 cm and 5 cm? Wait, no, let’s recheck. Wait, the diagonals here: one diagonal is split into 4 cm and 4 cm? No, wait the original problem: (2) has diagonals with segments 4 cm and 5 cm? Wait, no, maybe I misread. Wait, the key is:

• Rectangle: diagonals equal (so both diagonals same length, bisecting).
• Rhombus: diagonals perpendicular, bisecting, sides equal.
• Square: diagonals equal, perpendicular, bisecting, sides equal.

Wait, let's re-express:

• Parallelogram: Diagonals bisect each other (any length, not necessarily equal or perpendicular).
• Rectangle: Diagonals are equal (so each diagonal’s total length is equal) and bisect each other.
• Rhombus: Diagonals are perpendicular (intersect at 90°) and bisect each other (so all four segments from intersection are equal? No, rhombus diagonals bisect each other, but not necessarily equal in length. Wait, rhombus diagonals are perpendicular and bisect each other, but their lengths can differ. Wait, no: in a rhombus, diagonals are perpendicular bisectors, so the four triangles formed are congruent right triangles.
• Square: Diagonals are equal, perpendicular, and bisect each other (so all four segments are equal, and diagonals equal in length).

Figure (1)

Diagonals: segments 2 cm, 2 cm, 6 cm, 6 cm. So diagonals bisect each other (2+6=8 cm total for each diagonal? Wait, no—wait, the diagonals cross at the midpoint, so each diagonal is split into two equal parts. Wait, no: in a parallelogram, diagonals bisect each other, so each diagonal is split into two equal segments. Wait, the diagram (1) has diagonals with segments 2 cm and 6 cm? That can’t be—wait, no, maybe the diagonals are 4 cm (2+2) and 12 cm (6+6)? No, that doesn’t make sense. Wait, maybe the diagonals are 4 cm (2+2) and 12 cm (6+6)? No, that’s not right. Wait, maybe the problem is that (1) has diagonals that bisect each other (so it’s a parallelogram), (2) has diagonals equal (so rectangle), (3) has diagonals perpendicular (rhombus), (4) has diagonals equal and perpendicular (square). Wait, let's try again:

• Figure (1): Diagonals bisect each other (segments 2 cm and 6 cm), so total diagonals are 4 cm and 12 cm? No, that can’t be. Wait, no—if diagonals bisect each other, then each diagonal is split into two equal parts. So if one diagonal has segments 2 cm and 2 cm, that diagonal is 4 cm. The other diagonal has segments 6 cm and 6 cm, so 12 cm. So diagonals are not equal, not perpendicular. So this is a parallelogram.

• Figure (2): Diagonals bisect each other (segments 4 cm and 5 cm? Wait, no, the diagram shows 4 cm and 5 cm? Wait, maybe the diagonals are equal in length? Wait, maybe (2) has diagonals with total length equal (e.g., each diagonal is 8 cm? No, maybe the segments are 4 cm and 4 cm, 5 cm and 5 cm? Wait, the problem says “choose either parallelogram, rectangle, rhombus, or square”. Let's use the properties:

• Rectangle: Diagonals are equal (so both diagonals same length) and bisect each other. So if diagonals are equal, it’s a rectangle.
• Rhombus: Diagonals are perpendicular (intersect at 90°) and bisect each other.
• Square: Diagonals are equal, perpendicular, and bisect each other.

Figure (3)

Diagonals are perpendicular (right angle at intersection) and bisect each other (all four segments 4 cm). So this is a rhombus (perpendicular diagonals, bisecting, sides equal).

Figure (4)

Diagonals are equal (all segments 5 cm, so diagonals are 10 cm each) and bisect each other, and perpendicular? Wait, no—if all segments are 5 cm, and diagonals are equal, then it’s a square? Wait, no—if diagonals are equal and bisect each other, it’s a rectangle. But if also perpendicular, it’s a square. Wait, the segments are 5 cm each, so diagonals are equal (5+5=10 cm), bisect each other, and if they are perpendicular, it’s a square. But maybe (4) has diagonals equal (so rectangle) or square. Wait, let's assign:

• (1): Parallelogram (diagonals bisect, not equal/perpendicular)
• (2): Rectangle (diagonals equal, bisecting)
• (3): Rhombus (diagonals perpendicular, bisecting)
• (4): Square (diagonals equal, perpendicular, bisecting, all segments equal)

Wait, but the problem says “use each shape only once: parallelogram, rectangle, rhombus, square”. So:

• (1): Parallelogram
• (2): Rectangle (diagonals equal, bisecting)
• (3): Rhombus (perpendicular diagonals, bisecting)
• (4): Square (diagonals equal, perpendicular, bisecting, all sides implied equal)

Question 4 (True/False on Quadrilateral Properties)

(1) Squares are rhombuses.

A rhombus is a quadrilateral with all sides equal and diagonals perpendicular bisectors. A square has all sides equal and diagonals perpendicular bisectors. So square is a special case of rhombus. True (○).

(2) Rhombuses are rectangles.

A rectangle has all angles 90° and diagonals equal. A rhombus has all sides equal, but angles are not necessarily 90° (unless it’s a square). So rhombuses are not rectangles (unless square). False (×).

(3) Rectangles are squares.

A square has all sides equal and all angles 90°. A rectangle has all angles 90° but sides not necessarily equal (unless it’s a square). So rectangles are not squares (unless square). False (×).

(4) Both rectangles and rhombuses are parallelograms.

A parallelogram has opposite sides parallel and equal, diagonals bisect each other. A rectangle has opposite sides parallel/equal, angles 90° (so parallelogram). A rhombus has opposite sides parallel/equal, all sides equal (so parallelogram). True (○).

(5) Squares are rectangles, rhombuses, and parallelograms.

• Square is a rectangle (all angles 90°, diagonals equal, bisecting).
• Square is a rhombus (all sides equal, diagonals perpendicular bisectors).
• Square is a parallelogram (opposite sides parallel/equal, diagonals bisecting).

True (○).

(6) Quadrilaterals that are neither rectangles nor rhombuses are not parallelograms.

A parallelogram can be neither a rectangle nor a rhombus (e.g., a slanted parallelogram with unequal sides and non - 90° angles). So a quadrilateral could be a parallelogram without being a rectangle or rhombus. Thus, the statement is false. False (×).

Final Answers (Question 4)

(1) $\boldsymbol{\bigcirc}$
(2) $\boldsymbol{\times}$
(3) $\boldsymbol{\times}$
(4) $\boldsymbol{\bigcirc}$
(5) $\boldsymbol{\bigcirc}$
(6) $\boldsymbol{\times}$

Question 3 (Diagonals to Shapes)

• (1): Parallelogram
• (2): Rectangle (diagonals equal, bisecting)
• (3): Rhombus (perpendicular diagonals, bisecting)
• (4): Square (diagonals equal, perpendicular, bisecting, all segments equal)

(Note: For Question 3, the exact diagram details are a bit unclear, but the above is the logical assignment based on properties.)