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Categories of quadrilaterals

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5405073
Which named quadrilateral can have \(4\) equal sides without having \(4\) square corners?

Hints

- Focus on the attribute about equal side lengths. - Remember that one category does not require square corners.

Solution

1. A quadrilateral with \(4\) equal sides is a rhombus. 2. A rhombus does not have to have square corners. 3. When it does not have \(4\) square corners, it is a non-square rhombus.

Answer

A rhombus.
5404703
Lena says, “A square is not a quadrilateral because it has its own name.” Is Lena correct? Explain.

Hints

- Think about the meaning of the word quadrilateral. - Check whether a square has the attribute needed for that larger category.

Solution

1. A quadrilateral is any closed shape with \(4\) straight sides. 2. A square has \(4\) straight sides. 3. Therefore, a square is a quadrilateral, so Lena is not correct.

Answer

No. A square has \(4\) straight sides, so it is a quadrilateral.
5404753
A quadrilateral has \(4\) equal sides, but its corners are not all square corners. Is it a square, a non-square rhombus, or a rectangle that is not a square?

Hints

- Start with the clue about the side lengths. - Then use the corner clue to decide whether the shape is a square.

Solution

1. Four equal sides make the quadrilateral a rhombus. 2. A square must also have \(4\) square corners. 3. Because the corners are not all square corners, the shape is a non-square rhombus.

Answer

It is a non-square rhombus.
5404803
A quadrilateral has \(4\) equal sides and \(4\) square corners. Which is the most specific name: quadrilateral, rectangle, rhombus, or square?

Hints

- Use both the side clue and the corner clue. - Choose the name that includes all the given attributes, not only one of them.

Solution

1. Four equal sides make the shape a rhombus. 2. Four square corners make the shape a rectangle. 3. A quadrilateral with both attributes is a square. 4. Square is the most specific name.

Answer

The most specific name is square.
5404863
Which of these are quadrilaterals? Choose all that apply. A. A square B. A triangle C. A rectangle D. A closed shape with \(4\) straight sides and no equal sides E. A pentagon

Hints

- Use the defining number of straight sides for a quadrilateral. - Test every choice, including shapes with more specific names.

Solution

1. A square has \(4\) straight sides, so A is a quadrilateral. 2. A triangle has \(3\) sides, so B is not. 3. A rectangle has \(4\) straight sides, so C is a quadrilateral. 4. Choice D directly describes a quadrilateral. 5. A pentagon has \(5\) sides, so E is not.

Answer

A, C, and D.
5404893
Which single clue guarantees that a quadrilateral is a rectangle? A. It has \(4\) sides. B. It has \(4\) equal sides. C. It has \(4\) square corners.

Hints

- Ask which clue separates rectangles from other quadrilaterals. - Test whether each clue could also describe a non-rectangle.

Solution

1. Every quadrilateral has \(4\) sides, so A is not enough. 2. Four equal sides guarantee a rhombus, but not necessarily a rectangle. 3. Four square corners guarantee a rectangle. 4. Choice C is correct.

Answer

C. It has \(4\) square corners.
5404973
A rectangle has all \(4\) sides equal. What more specific quadrilateral is it?

Hints

- Keep the corner attributes that come with the word rectangle. - Combine those corners with the new information about all four sides.

Solution

1. A rectangle has \(4\) square corners. 2. The shape also has \(4\) equal sides. 3. A quadrilateral with both attributes is a square.

Answer

It is a square.
5405003
A quadrilateral has \(4\) square corners. Its side lengths, in order around the shape, are \(6, 3, 6, 3\). Choose the most specific description: square; rectangle, but not a square; or rhombus, but not a square.

Hints

- Use the corner clue to identify the main category. - Compare all four side lengths. - A square needs both four square corners and four equal sides.

Solution

1. Four square corners make the quadrilateral a rectangle. 2. The side lengths are not all equal because \(6 \ne 3\). 3. Therefore, the shape is a rectangle, but not a square.

Answer

It is a rectangle, but not a square.
5405043
Diego says, “A square cannot be a rhombus because a square has square corners.” Is Diego correct? Explain.

Hints

- Use the side-length attribute that defines a rhombus. - Check whether a square has that attribute. - Do not assume that one category name removes another valid name.

Solution

1. A rhombus is a quadrilateral with \(4\) equal sides. 2. A square has \(4\) equal sides. 3. Square corners do not prevent a shape from being a rhombus. 4. Therefore, Diego is not correct.

Answer

No. A square has \(4\) equal sides, so it is also a rhombus.
5405113
A sorting tray has four bins: square, rectangle that is not a square, rhombus that is not a square, and other quadrilateral. A shape has \(4\) equal sides and at least one corner that is not square. Which bin should receive it?

Hints

- Use the side-length clue to choose a main category. - Use the corner clue to decide between the square and non-square bins.

Solution

1. Four equal sides make the shape a rhombus. 2. At least one corner is not square, so the shape is not a square. 3. It belongs in the “rhombus that is not a square” bin.

Answer

Rhombus that is not a square.
5405133
Which quadrilateral category follows this rule: “The shape has \(4\) square corners. Its sides may all be equal, or it may have two different side lengths”?

Hints

- Focus on the corner rule that every shape in the category shares. - Remember that the category may include squares as well as non-squares.

Solution

1. Every quadrilateral with \(4\) square corners is a rectangle. 2. A square is included because it is a rectangle with all sides equal. 3. Non-square rectangles are also included. 4. The category is rectangle.

Answer

The category is rectangle.
5405173
Which quadrilateral category follows this rule: “All \(4\) sides are equal. The corners may be square or may not be square”?

Hints

- Focus on the one attribute shared by every shape in the category. - Remember that the category can include a more specific shape.

Solution

1. Four equal sides define a rhombus. 2. Squares are included because they also have \(4\) equal sides. 3. Non-square rhombuses are included as well. 4. The category is rhombus.

Answer

The category is rhombus.
5405193
Mina says, “A rectangle can never have \(4\) equal sides.” Is Mina correct? Explain.

Hints

- Think about whether squares are included in the rectangle category. - Check the corner and side attributes of a square.

Solution

1. A rectangle is defined by its \(4\) square corners. 2. Some rectangles also have \(4\) equal sides. 3. Those rectangles are squares. 4. Therefore, Mina is not correct.

Answer

No. A square is a rectangle with \(4\) equal sides.
5405323
A quadrilateral has \(4\) equal sides and is not a rectangle. Choose its most specific description: square; rhombus, but not a square; or rectangle, but not a square.

Hints

- Use the four-equal-sides clue first. - A square is also a rectangle. - Use “not a rectangle” to rule out the square choice.

Solution

1. Four equal sides make the quadrilateral a rhombus. 2. Since it is not a rectangle, it does not have \(4\) square corners. 3. Therefore, it is a rhombus, but not a square.

Answer

Rhombus, but not a square.
5405353
A square and a rhombus that is not a square both have \(4\) equal sides. Which corner attribute tells them apart?

Hints

- The side lengths are already the same in both descriptions. - Compare the corners instead.

Solution

1. A square has \(4\) square corners. 2. A non-square rhombus does not have all \(4\) corners square. 3. The distinguishing attribute is whether all four corners are square.

Answer

A square has \(4\) square corners; a non-square rhombus does not.
5405653
Each grid square represents \(1\) square unit. Use the figure. Choose the most specific category for the combined outside shape: square; rectangle, but not a square; or rhombus, but not a square. Explain.
Figure for problem 540565

Hints

- Trace only the outside boundary. - Use the grid to compare a horizontal side with a vertical side. - Check the corner attribute before choosing the most specific category.

Solution

1. The inside seam is not part of the outside boundary. 2. The outside shape has \(4\) square corners, so it is a rectangle. 3. Its horizontal sides are \(4\) units long and its vertical sides are \(2\) units long, so all \(4\) sides are not equal. 4. Therefore, the most specific category is rectangle, but not a square.

Answer

Rectangle, but not a square.
5404833
Classify a) and b) using the phrase bank: “square,” “rhombus, but not a square,” or “rectangle, but not a square.” Explain which attributes you used.
Figure for problem 540483

Hints

- For shape a), compare all four side lengths on the grid. - For shape b), check the corners and compare adjacent side lengths. - A square must have both \(4\) equal sides and \(4\) square corners.

Solution

1. Shape a) has \(4\) equal sides, but its corners are not square corners. It is a rhombus, but not a square. 2. Shape b) has \(4\) square corners and two different side lengths. It is a rectangle, but not a square.

Answer

a) Rhombus, but not a square b) Rectangle, but not a square
5404933
A quadrilateral has \(4\) equal sides. What can you know for sure? Choose all that apply. A. It is a rhombus. B. It is a quadrilateral. C. It is a square. D. It is a rectangle.

Hints

- Separate what is given from what is not given. - Ask whether equal sides alone tell you anything about the corners. - Choose only statements that must be true.

Solution

1. Four equal sides guarantee that the shape is a rhombus, so A is true. 2. The shape is already stated to be a quadrilateral, so B is true. 3. The corner information is missing, so the shape may or may not be a square or rectangle. 4. C and D are not guaranteed.

Answer

A and B.
5405093
Use this phrase bank to name each quadrilateral as specifically as possible: “square,” “rhombus, but not a square,” and “rectangle, but not a square.” a) Four equal sides and four square corners b) Four equal sides and at least one corner not square c) Four square corners and two different side lengths

Hints

- For each part, begin with the side clue or the corner clue. - Then use the second clue to decide whether the shape can be a square. - Choose one exact phrase from the bank for each part.

Solution

1. a) Both defining attributes together make a square. 2. b) Four equal sides make a rhombus, and the corner clue shows it is not a square. 3. c) Four square corners make a rectangle, and the different side lengths show it is not a square.

Answer

a) Square b) Rhombus, but not a square c) Rectangle, but not a square
5405223
A quadrilateral has \(4\) square corners, but its side lengths are not shown. Which descriptions are possible? Choose from “square,” “rectangle, but not a square,” and “rhombus, but not a square.” Explain.

Hints

- Use the corner clue to identify the guaranteed broad category. - Consider one case in which all four sides are equal and one case in which they are not. - Check each phrase from the bank against the square-corner clue.

Solution

1. Four square corners guarantee that the shape is a rectangle. 2. If all \(4\) sides are equal, the shape is a square. 3. If the side lengths are not all equal, the shape is a rectangle, but not a square. 4. A rhombus that is not a square cannot have four square corners, so that description is not possible.

Answer

The possible descriptions are “square” and “rectangle, but not a square.”
5405243
A sorting report for one quadrilateral says rectangle: no, rhombus: no, and square: yes. Which entries must be corrected?

Hints

- Start with the most specific category marked yes. - Check which larger or overlapping categories every square must belong to. - Correct every entry that conflicts with those required memberships.

Solution

1. A square has \(4\) square corners, so every square is a rectangle. 2. A square has \(4\) equal sides, so every square is a rhombus. 3. If square is yes, rectangle and rhombus must also be yes. 4. Therefore, both no entries must be changed to yes.

Answer

Change rectangle and rhombus from no to yes.
5405283
A rectangle bin includes every rectangle, including squares. Which descriptions belong in the bin? A. Four square corners and side lengths \(2, 5, 2, 5\) B. Four equal sides and four square corners C. Four equal sides and at least one corner not square D. Four sides with no square-corner information

Hints

- Use the corner requirement for the rectangle category. - Remember that squares are included. - Reject descriptions that do not guarantee all four square corners.

Solution

1. A has \(4\) square corners, so it is a rectangle. 2. B has \(4\) square corners, so it is a square and also a rectangle. 3. C is a non-square rhombus, not a rectangle. 4. D does not guarantee a rectangle. 5. A and B belong in the bin.

Answer

A and B.
5405303
A rhombus bin includes every rhombus, including squares. Which descriptions belong in the bin? A. Four equal sides and four square corners B. Four equal sides and at least one corner not square C. Four square corners and side lengths \(2, 5, 2, 5\) D. Four sides with no equal-side information

Hints

- Use the side-length requirement for every rhombus. - Remember that squares are included in the rhombus category. - Do not assume equal sides when the description does not say so.

Solution

1. A has \(4\) equal sides, so it is a square and also a rhombus. 2. B has \(4\) equal sides, so it is a non-square rhombus. 3. C is a non-square rectangle and does not have \(4\) equal sides. 4. D does not guarantee a rhombus. 5. A and B belong in the bin.

Answer

A and B.
5405363
A quadrilateral has \(4\) square corners and side lengths \(4, 6, 4, 6\). Is it a rectangle, a rhombus, both, or neither?

Hints

- Use the corner clue for one category. - Use the side-length list for the other category. - Check whether both definitions are satisfied.

Solution

1. Four square corners make the shape a rectangle. 2. The side lengths are not all equal because \(4 \ne 6\). 3. Therefore, it is not a rhombus. 4. It is a rectangle only.

Answer

It is a rectangle only.
5405443
A quadrilateral is in the rectangle category but not in the rhombus category. What must be true about its side lengths?

Hints

- Think about the side attribute required for every rhombus. - Use the category that does not fit. - Decide what must be false about the four side lengths.

Solution

1. A rectangle has \(4\) square corners. 2. If all \(4\) sides were equal, the rectangle would also be a rhombus and therefore a square. 3. Because the rhombus category does not fit, the side lengths cannot all be equal.

Answer

Its \(4\) side lengths are not all equal.
5405503
Exactly one of these two category labels fits a mystery quadrilateral: rectangle or rhombus. The shape has all \(4\) sides equal. Which label fits, and what does that tell you about its corners?

Hints

- Start with the clue about all four side lengths. - Use the words “exactly one” to rule out the other category. - State what must fail about the corners when the rectangle label does not fit.

Solution

1. Four equal sides make the shape a rhombus. 2. The problem says exactly one of the two labels fits, so rectangle does not fit. 3. A rectangle must have all \(4\) square corners. 4. Therefore, at least one corner of the rhombus is not square.

Answer

The rhombus label fits. At least one corner is not square.
5405563
A quadrilateral has \(4\) equal sides. Can you decide from this information alone whether it is a rectangle? Explain.

Hints

- Identify the category guaranteed by the side lengths. - Think about the extra attribute required for a rectangle. - Consider two possible corner arrangements.

Solution

1. Four equal sides make the quadrilateral a rhombus. 2. If all \(4\) corners are square, the rhombus is also a rectangle and a square. 3. If the corners are not all square, the rhombus is not a rectangle. 4. The side information alone does not determine the answer.

Answer

No. It is a rectangle only if all \(4\) corners are square.
5405583
Classify each quadrilateral as square, rectangle that is not a square, rhombus that is not a square, or other quadrilateral. <table><tr><th>Shape</th><th>All sides equal?</th><th>All corners square?</th></tr><tr><td>W</td><td>yes</td><td>yes</td></tr><tr><td>X</td><td>no</td><td>yes</td></tr><tr><td>Y</td><td>yes</td><td>no</td></tr><tr><td>Z</td><td>no</td><td>no</td></tr></table>

Hints

- Read each row as a pair of yes-or-no attributes. - Use equal sides to test the rhombus category. - Use square corners to test the rectangle category, then combine the results.

Solution

1. W has all equal sides and all square corners, so W is a square. 2. X has all square corners but not all equal sides, so X is a rectangle that is not a square. 3. Y has all equal sides but not all square corners, so Y is a rhombus that is not a square. 4. Z has neither special attribute, so Z belongs in the “other quadrilateral” category.

Answer

W: square X: rectangle that is not a square Y: rhombus that is not a square Z: other quadrilateral
5405083
A shape is known to be a quadrilateral. Is either clue alone enough to prove it is a square? a) All \(4\) sides are equal. b) All \(4\) corners are square. Explain.

Hints

- For each clue, imagine a quadrilateral that has that attribute but is not a square. - Decide what additional information is missing. - Compare what each clue guarantees by itself.

Solution

1. Clue a) alone proves the shape is a rhombus, but it could be a non-square rhombus. 2. Clue b) alone proves the shape is a rectangle, but it could be a non-square rectangle. 3. Neither clue alone proves the shape is a square. Both clues together would.

Answer

a) No. b) No. Both clues together are needed to prove the quadrilateral is a square.
5405403
Three quadrilaterals are named A, B, and C. Exactly one is a square, one is a rectangle that is not a square, and one is a rhombus that is not a square. - A and B each have all \(4\) sides equal. - B and C each have all \(4\) corners square. Classify A, B, and C.

Hints

- Look for the shape that appears in both attribute clues. - After identifying the square, use the remaining side clue and corner clue. - Each of the three category names is used exactly once.

Solution

1. B has all \(4\) sides equal and all \(4\) corners square, so B is the square. 2. A has all \(4\) sides equal, but it is not the square, so A is the rhombus that is not a square. 3. C has all \(4\) square corners, but it is not the square, so C is the rectangle that is not a square.

Answer

A is the rhombus that is not a square. B is the square. C is the rectangle that is not a square.
5405433
Owen says, “Any rhombus with perimeter \(20\) units must be a square.” Give a quadrilateral description that proves Owen is wrong.

Hints

- Use the perimeter to find the equal side length. - Keep the rhombus side condition true. - Change the corner condition needed for a square.

Solution

1. A rhombus with perimeter \(20\) units has four equal sides of \(20 \div 4 = 5\) units. 2. Choose a quadrilateral with side lengths \(5, 5, 5, 5\) and at least one corner that is not square. 3. The shape is a rhombus because all sides are equal, but it is not a square because not all corners are square. 4. This description is a counterexample to Owen's claim.

Answer

A rhombus with four sides of \(5\) units and at least one corner that is not square.
5405533
A class rule says, “A rectangle has \(4\) square corners and two long sides and two short sides.” Why is this rule incomplete? Rewrite it so it includes every rectangle.

Hints

- Test the rule with a square. - Decide whether side lengths must be different for every rectangle. - Keep only the attributes shared by all rectangles.

Solution

1. The rule excludes squares because a square does not have two long sides and two short sides. 2. A square still has \(4\) square corners, so it is a rectangle. 3. A complete rule is: A rectangle is a quadrilateral with \(4\) square corners.

Answer

The rule leaves out squares. A complete rule is: “A rectangle is a quadrilateral with \(4\) square corners.”
5405673
A rectangle has perimeter \(24\) units. Can the perimeter alone tell whether the rectangle is a square? Give one possible square side-length list and one possible non-square rectangle side-length list.

Hints

- Find four equal side lengths that total \(24\). - Find two different side lengths that repeat as opposite sides and also total \(24\). - Compare the categories of the two examples.

Solution

1. A square with side lengths \(6, 6, 6, 6\) has perimeter \(6 + 6 + 6 + 6 = 24\) units. 2. A non-square rectangle with side lengths \(4, 8, 4, 8\) has perimeter \(4 + 8 + 4 + 8 = 24\) units. 3. The same perimeter can belong to both categories. 4. Therefore, perimeter alone does not determine whether the rectangle is a square.

Answer

No. For example, \(6, 6, 6, 6\) makes a square, while \(4, 8, 4, 8\) makes a rectangle that is not a square; both have perimeter \(24\) units.

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