64. abstract reasoning order the terms in a diagram so that each term builds off the previous term(s). explain why each figure is in the location you chose. quadrilateral square rectangle rhombus parallelogram critical thinking in exercises 65–70, complete each statement with always, sometimes, or never. explain your reasoning. 65. a square is ______ a rhombus. 66. a rectangle is ______ a square. 67. a rectangle ______ has congruent diagonals. 68. the diagonals of a square ______ bisect its angles. 69. a rhombus ______ has four congruent angles. 70. a rectangle ______ has perpendicular diagonals.

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Accepted Answer

### 64. Abstract Reasoning (Ordering the Terms)
# Explanation:
## Step1: Identify the broadest term
The broadest term here is "quadrilateral" because a quadrilateral is defined as a four - sided polygon. All the other terms (parallelogram, rectangle, rhombus, square) are types of quadrilaterals. So we start with "quadrilateral".
## Step2: Next, identify the next level of classification
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. So "parallelogram" comes after "quadrilateral" since all parallelograms are quadrilaterals, but not all quadrilaterals are parallelograms.
## Step3: Identify the sub - types of parallelogram
A rectangle is a parallelogram with four right angles, and a rhombus is a parallelogram with four congruent sides. So "rectangle" and "rhombus" come after "parallelogram" as they are specific types of parallelograms.
## Step4: Identify the most specific term
A square is a special case of both a rectangle (it has four right angles) and a rhombus (it has four congruent sides). So a square is a sub - type of both rectangle and rhombus. So the order from broadest to most specific is: quadrilateral → parallelogram → rectangle, rhombus → square.

Explanation for each location:
- Quadrilateral: It is the most general term as it only requires a four - sided figure. All the other terms represent four - sided figures with additional properties.
- Parallelogram: A parallelogram has the property of opposite sides parallel, which is an additional property on top of being a quadrilateral. So all parallelograms are quadrilaterals, but not all quadrilaterals are parallelograms.
- Rectangle: A rectangle is a parallelogram with four right angles. So it has all the properties of a parallelogram plus the property of right angles.
- Rhombus: A rhombus is a parallelogram with four congruent sides. So it has all the properties of a parallelogram plus the property of congruent sides.
- Square: A square has four right angles (like a rectangle) and four congruent sides (like a rhombus). So it has all the properties of a rectangle and a rhombus, making it the most specific term.

# Answer:
The order from broadest to most specific is: quadrilateral → parallelogram → rectangle, rhombus → square.

### 65. A square is ______ a rhombus.
# Brief Explanations:
A rhombus is a quadrilateral with four congruent sides. A square has four congruent sides (and also four right angles). Since a square satisfies the definition of a rhombus (four congruent sides), a square is always a rhombus.
# Answer:
always

### 66. A rectangle is ______ a square.
# Brief Explanations:
A rectangle is a quadrilateral with four right angles. A square is a rectangle with the additional property that all four sides are congruent. A rectangle can have sides of different lengths (e.g., a rectangle with length 5 and width 3), in which case it is not a square. But if a rectangle has all four sides congruent, then it is a square. So a rectangle is sometimes a square.
# Answer:
sometimes

### 67. A rectangle ______ has congruent diagonals.
# Brief Explanations:
In a rectangle, we can prove that the diagonals are congruent. Consider a rectangle $ABCD$ with $AB = CD$, $AD=BC$ and $\angle ABC=\angle BAD = 90^{\circ}$. By the Pythagorean theorem, the length of diagonal $AC=\sqrt{AB^{2}+BC^{2}}$ and the length of diagonal $BD=\sqrt{AB^{2}+AD^{2}}$. But since $AD = BC$ (opposite sides of a rectangle are equal), $AC = BD$. So a rectangle always has congruent diagonals.
# Answer:
always

### 68. The diagonals of a square ______ bisect its angles.
# Brief Explanations:
In a square, each angle is $90^{\circ}$. The diagonals of a square divide each $90^{\circ}$ angle into two $45^{\circ}$ angles. For example, in square $ABCD$ with diagonal $AC$, $\triangle ABC$ is an isosceles right triangle, and the diagonal $AC$ bisects $\angle BAD$ and $\angle BCD$ (and similarly for the other diagonal). So the diagonals of a square always bisect its angles.
# Answer:
always

### 69. A rhombus ______ has four congruent angles.
# Brief Explanations:
A rhombus has four congruent sides. However, the angles of a rhombus can be non - right angles (e.g., a rhombus with angles $60^{\circ}$ and $120^{\circ}$). A rhombus will have four congruent angles only if it is a square (a special case of a rhombus). So a rhombus sometimes has four congruent angles.
# Answer:
sometimes

### 70. A rectangle ______ has perpendicular diagonals.
# Brief Explanations:
In a rectangle, the diagonals are congruent but not necessarily perpendicular. A rectangle will have perpendicular diagonals only if it is a square (a special case of a rectangle). For example, in a rectangle with length $l$ and width $w$ where $l
eq w$, the diagonals are not perpendicular. So a rectangle sometimes has perpendicular diagonals.
# Answer:
sometimes

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Question

Explanation:

64. Abstract Reasoning (Ordering the Terms)

Step1: Identify the broadest term

Step2: Next, identify the next level of classification

Step3: Identify the sub - types of parallelogram

Step4: Identify the most specific term

Answer:

65. A square is ______ a rhombus.