- Check the side lengths of each shape first. - Then check whether all four corners are square corners. - The correct shape must satisfy both instructions.

1. Shape a) has \(4\) equal sides and \(4\) square corners. 2. Shape b) has \(4\) equal sides, but its corners are not square corners. 3. Shape c) has \(4\) square corners, but not all \(4\) sides are equal. 4. Therefore, shape a) follows both instructions.

Shape a).

- The app already guarantees the square-corner attribute. - Decide what must be different so the result is not a square. - Compare the two numbers in each pair.

1. A square has equal length and width. 2. A rectangle that is not a square needs two different side lengths. 3. Only \(2\) and \(6\) are different, so that pair works.

Use side lengths \(2\) and \(6\).

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

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.

A rhombus.

- Focus on the corner attribute. - Recall which quadrilateral category is defined by four square corners.

1. A rectangle is a quadrilateral with \(4\) square corners. 2. The side lengths do not need to be known to guarantee the rectangle category.

Rectangle.

- A square needs four equal sides. - Use the given side length for every side. - Recall the corner attribute of a square.

1. A square has four equal sides, so every side is \(3\,\text{units}\) long. 2. A square also has four square corners.

\(3\,\text{units}, 3\,\text{units}, 3\,\text{units}, 3\,\text{units}\), with four square corners.

- Look at the steps from one peg to the next along each slanted side. - Compare the corners with a square corner. What does that rule out?

1. All four sides have the same length on the peg grid. 2. The corners are not right angles, so it cannot be a square or rectangle. 3. A quadrilateral with four equal sides is a rhombus.

Rhombus that is not a square. All four sides are equal, but the corners are not right angles.

- Check each corner against the corner of a square grid cell. - Name the quadrilateral with that corner property.

1. Each of the four corners is a right angle, so there are \(4\) right angles. 2. A quadrilateral with \(4\) right angles is a rectangle.

\(4\) right angles. The shape is a rectangle.

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

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.

No. A square has \(4\) straight sides, so it is a quadrilateral.

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

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.

It is a non-square rhombus.

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

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.

A, C, and D.

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

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.

C. It has \(4\) square corners.

- The current instructions already make one quadrilateral category. - Think about the extra side attribute that makes this category more specific.

1. The selected instructions guarantee a rectangle. 2. To make that rectangle a square, all \(4\) sides must be equal. 3. Add the instruction “all sides equal.”

Add “all sides equal.”

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

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

It is a square.

- 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.

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.

It is a rectangle, but not a square.

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

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.

The category is rectangle.

- Identify the side-length feature that every square has. - Look for the choice that makes that feature impossible. - A condition already true for all rectangles does not rule out squares.

1. A square has all \(4\) sides equal. 2. If two adjacent sides have different lengths, the rectangle cannot be a square, so A works. 3. Choice B forces a square. Choice C is true for every rectangle, including squares, so it does not rule out a square.

A. Make two adjacent sides different lengths.

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

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.

The category is rhombus.

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

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.

Rhombus, but not a square.

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

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.

A square has \(4\) square corners; a non-square rhombus does not.

- What would your finger trace if you went all the way around the outside? - Compare the length of a horizontal outside side with a vertical one. - Think about which corners a rectangle needs.

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 \(4\,\text{units}\) long and its vertical sides are \(2\,\text{units}\) long. The sides are not all equal. 4. Therefore, it is a rectangle, but not a square.

Rectangle, but not a square.

- Look at the corners of each figure. Are they all square corners? - Compare the lengths of the four sides in each figure.

1. Figure a) has four right angles, so it is a rectangle. 2. Figure b) has four equal sides, so it is a rhombus. 3. Figure c) has four sides, but they are not all equal and its corners are not all right angles. 4. Figure c) is a quadrilateral that is neither a rectangle nor a rhombus.

Figure c). It has four straight sides, but its sides are not all equal and its corners are not all right angles.

- Look for four square corners first. - Then compare adjacent side lengths. - A non-square rectangle must have two different side lengths.

1. Figure a) has four square corners and two different side lengths, so it is a rectangle but not a square. 2. Figure b) has four equal sides and four square corners, so it is a square. 3. Figure c) does not have four square corners.

Figure a).

- Start with what would make a rhombus become a square. - Use the words “not a square” to rule out one corner condition.

1. A rhombus has four equal sides. 2. If all four corners were square, the rhombus would also be a square. 3. Because it is not a square, at least one corner is not square.

At least one corner is not square.

- Check equal sides and square corners separately. - Combine those two yes-or-no attributes for each figure. - Use the most specific phrase that fits.

1. Figure a) has four equal sides and four square corners, so it is a square. 2. Figure b) has four square corners but two different side lengths, so it is a rectangle but not a square. 3. Figure c) has four equal sides but not four square corners, so it is a rhombus but not a square. 4. Figure d) has four sides, but neither four equal sides nor four square corners. It is neither a rectangle nor a rhombus.

a) Square b) Rectangle but not a square c) Rhombus but not a square d) Quadrilateral that is neither a rectangle nor a rhombus

- Opposite sides of a rectangle have matching lengths. - Keep the two adjacent lengths in the order given. - Remember the corner condition that every rectangle satisfies.

1. Opposite sides of a rectangle have the same length. 2. Starting with adjacent sides of \(5\,\text{units}\) and \(2\,\text{units}\), the four lengths are \(5, 2, 5, 2\) units in order. 3. A rectangle has four square corners. 4. Because neighboring sides differ in length, this rectangle is not a square.

\(5\,\text{units}, 2\,\text{units}, 5\,\text{units}, 2\,\text{units}\), with four square corners.

- Keep the defining side attribute of a rhombus. - Compare that with the extra corner attribute of a square. - State the corner condition that prevents the rhombus from being a square.

1. A rhombus has four equal sides. 2. A square has four equal sides and four square corners. 3. Therefore, a rhombus that is not a square must keep the four equal sides and have at least one corner that is not square.

Four equal sides and at least one corner that is not square.

- The direction a figure faces does not decide its name. - Compare neighboring side lengths on the peg grid. - Check for four square corners.

1. The figure has four square corners, so it is a rectangle. 2. One pair of opposite sides is longer than the other, so not all four sides are equal. 3. It is a rectangle but not a square. Tilting a figure does not change its shape category.

Rectangle but not a square.

- Compare the lengths of neighboring sides in each figure. - Look closely at how the sides meet at each corner. - Which two properties must a square have together?

1. In a), the four corners are right angles, but the sides are not all the same length. It is a rectangle but not a square. 2. In b), all four sides have the same length, but the corners are not right angles. It is a rhombus but not a square.

a) Rectangle but not a square: four right angles, but not four equal sides. b) Rhombus but not a square: four equal sides, but not four right angles.

- Check each corner, not just the ones along the bottom. - How many right angles must a rectangle have?

1. The bottom-right corner is a right angle because a horizontal side meets a vertical side. 2. The corner at the top-right is not a right angle. 3. A rectangle needs four right angles. One right angle is not enough.

No. It has only \(1\) right angle, but a rectangle has \(4\).

- Count the spaces between pegs along each horizontal edge. - Compare each corner with a square corner on the grid. - What must be true about both the sides and corners of a rectangle?

1. The top side measures \(3\) grid units and the bottom side measures \(6\) grid units. 2. None of the four corners is a right angle. 3. A rectangle has equal opposite sides and four right angles, so this figure is not a rectangle.

The top is \(3\) grid units long, and the bottom is \(6\). The opposite sides are not equal, and there are \(0\) right angles. Both facts show that it is not a rectangle.

- Use the grid to read each side length of 2). - Check whether the corners of 2) are square corners. - Compare the equality of the four sides in both figures.

1. In 2), the top and bottom sides are \(7\) units long, and the other two sides are \(4\) units long. The list is \(7, 4, 7, 4\). 2. Figure 2) has \(4\) square corners, so it is a rectangle. Its sides are not all equal, so it is not a square. 3. Both figures have \(4\) sides and \(4\) square corners. Unlike 1), 2) does not have \(4\) equal sides.

a) \(7, 4, 7, 4\) b) Rectangle, but not a square c) Both have \(4\) sides and \(4\) square corners, but only 1) has all \(4\) sides equal.

- 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.

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.

a) Rhombus, but not a square b) Rectangle, but not a square

- 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.

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.

A and B.

- In a four-square-corner shape, opposite sides must match. - In a side list written around the shape, compare the first with the third and the second with the fourth. - After finding a possible rectangle, check whether all four sides are equal.

1. In a rectangle, opposite sides must have equal lengths. In a list written around the shape, the first and third lengths must match, and the second and fourth lengths must match. 2. In A, the opposite pairs are \(7\) and \(7\), and \(3\) and \(3\). Because the two side lengths differ, this makes a rectangle that is not a square. 3. In B, the opposite pairs would be \(7\) and \(3\), so the sides cannot close to make a shape with four square corners. 4. In C, all four sides are equal, so the result would be a square.

A. List B cannot make a closed shape with four square corners, and list C makes a square.

- 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.

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.

a) No. b) No. Both clues together are needed to prove the quadrilateral is a square.

- 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.

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.

a) Square b) Rhombus, but not a square c) Rectangle, but not a square

- Start with what the four square corners guarantee. - Consider both equal and unequal side lengths. - Test each description against the corner clue.

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

The possible descriptions are “square” and “rectangle, but not a square.”

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

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.

A and B.

- 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.

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.

A and B.

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

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.

It is a rectangle only.

- Combine the rectangle corner rule with the equal-side instruction. - Identify the more specific shape that results. - Compare it with the “not a square” condition.

1. A rectangle with all \(4\) sides equal has \(4\) square corners and \(4\) equal sides. 2. Such a rectangle is a square. 3. Therefore, it cannot also be a rectangle that is not a square.

No. A rectangle with all four sides equal is a square.

- 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.

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.

Its \(4\) side lengths are not all equal.

- 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.

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.

The rhombus label fits. At least one corner is not square.

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

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.

No. It is a rectangle only if all \(4\) corners are square.

- 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.

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.

W: square X: rectangle that is not a square Y: rhombus that is not a square Z: other quadrilateral

- Separate side-length information from corner information. - Recall the defining corner attribute of a rectangle. - Decide whether the given clue guarantees that attribute.

1. A rectangle must have four square corners. 2. Equal opposite sides do not tell us that the corners are square. 3. Therefore, the given information is not enough to prove the quadrilateral is a rectangle.

No. You also need information that guarantees four square corners.

- Compare “one square corner” with the full rectangle requirement. - To disprove the statement, keep the given clue true while making the rectangle rule fail. - You only need one possible counterexample description.

1. A rectangle needs all four corners to be square. 2. A quadrilateral can have one square corner and another corner that is not square. 3. Such a quadrilateral is a counterexample, so Lila's statement is false.

No. A counterexample is any quadrilateral with one square corner but not all four corners square.

- Use each “not” statement separately. - Ask which defining attribute must fail for a rectangle. - Ask which defining attribute must fail for a rhombus.

1. If all four corners were square, the shape would be a rectangle, so A must be true. 2. If all four sides were equal, the shape would be a rhombus, so B must be true. 3. It is already a quadrilateral, so it has four straight sides; C and D are false.

A and B.

- Keep the same four vertices except for the one named in the instruction. - After the imagined move, count the outside sides. - Use the broad category determined by that side count.

1. Moving one vertex changes some side lengths and corner sizes. 2. The new figure still has four straight sides and remains closed. 3. Therefore, it is still a quadrilateral, although it is no longer a square.

The new figure is still a quadrilateral.

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

1. A rhombus with perimeter \(20\,\text{units}\) has four equal sides of \(20 \div 4 = 5\,\text{units}\). 2. Choose a quadrilateral with side lengths \(5\,\text{units}, 5\,\text{units}, 5\,\text{units}, 5\,\text{units}\) 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.

A rhombus with four sides of \(5\,\text{units}\) and at least one corner that is not square.

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

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.

The rule leaves out squares. A complete rule is: “A rectangle is a quadrilateral with \(4\) square corners.”

- Find four equal side lengths that total \(24\,\text{units}\). - Find two different side lengths that repeat as opposite sides and have the same total perimeter. - Compare the categories of the two examples.

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

No. For example, \(6\,\text{units}, 6\,\text{units}, 6\,\text{units}, 6\,\text{units}\) makes a square, while \(4\,\text{units}, 8\,\text{units}, 4\,\text{units}, 8\,\text{units}\) makes a rectangle that is not a square. Both have perimeter \(24\,\text{units}\).

- Find the shape that appears in both attribute groups. - Use the “exactly one square” clue before classifying the other two. - Then match the remaining corner and side attributes.

1. Q has both four square corners and four equal sides, so Q is the square. 2. P has four square corners but is not the square, so P is a rectangle but not a square. 3. R has four equal sides but is not the square, so R is a rhombus but not a square.

P: rectangle but not a square Q: square R: rhombus but not a square