Geometry B U2 L5 Conditions for Rhombuses, Rectangles, and Squares Answers -
This has two parts. The first will be the practice answers, the second part will be the quick check answers.
Practice Answers -
1. Which of the following statements are true? Statement 1: A parallelogram is always a rectangle. Statement 2: A square is always a rhombus. Statement 3: A rectangle is always a square.
Answer - C. Statement 2 only
2. Which of the following statements can be made about the parallelogram shown below? Note that the figure is not drawn to scale.
Answer - A. It is a rhombus.
3. For which value of x is the figure a rhombus?
Answer - C. x = 12
4. For which value of x is the figure a rectangle?
Answer - A. x = 1
5. A carpenter is building a bookcase. How can the carpenter use a tape measure to check if the bookshelf is rectangular?
Answer - B. She can measure the opposite sides. If they are congruent, then the bookshelf is in the shape of a parallelogram. Then she can measure the diagonals. If they are congruent, then the bookshelf is rectangular.
6. Suppose the diagonals of a parallelogram are both perpendicular and congruent. Which type of special quadrilateral is it? Use the most descriptive name possible.
Answer - C. square
7. For which value of x is the figure a rectangle?
Answer - B. x = 11
8. For which value of x is the figure a rhombus?
Answer - D. x = 7
9. Which reason completes the proof below? AB ≅ CF and BC ≅ AD - ?
Answer - A. Opposite sides of a parallelogram are congruent.
10. In the figure below, LN = 4x - 7 and MO = 2x + 13. For which value of x is the figure a rectangle?
Answer - A. x = 10
Quick Check Answers -
1. Which is the most precise name of this figure?
Answer - B. Rhombus
2. Which of the following conditions or set of conditions must be met for a parallelogram to be a rectangle?
Answer - B. Diagonals are congruent
3. Which of the following conditions or set of conditions is sufficient for a parallelogram to be a square
Answer - A. Diagonals are perpendicular ,and diagonals are congruent
4. For which of x is XYZA a rectangle?
Answer - C. 4
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Great! Here are the step-by-step explanations for each answer:
Practice Answers:
1. Statement 1: A parallelogram is always a rectangle. False, a parallelogram can be a rectangle but doesn't have to be.
Statement 2: A square is always a rhombus. True, a square is a special type of rhombus where all angles are right angles.
Statement 3: A rectangle is always a square. False, a rectangle can have different angles, while a square has all angles equal.
Answer: C. Statement 2 only
2. From the given figure, we can observe that the opposite sides are congruent. This is a property of rhombuses.
Answer: A. It is a rhombus.
3. The figure will be a rhombus when all sides are congruent. The given figure has x=12, which will make all sides congruent.
Answer: C. x = 12
4. The figure will be a rectangle when all angles are right angles. This occurs when x=1.
Answer: A. x = 1
5. To check if the bookshelf is rectangular, the carpenter can perform two measurements. First, she can measure the opposite sides. If they are congruent, then the bookshelf is in the shape of a parallelogram. Then, she can measure the diagonals. If they are congruent, then the bookshelf is a rectangle.
Answer: B. She can measure the opposite sides. If they are congruent, then the bookshelf is in the shape of a parallelogram. Then she can measure the diagonals. If they are congruent, then the bookshelf is rectangular.
6. If the diagonals of a parallelogram are both perpendicular and congruent, then it is a special type of quadrilateral called a square.
Answer: C. square
7. The figure will be a rectangle when all angles are right angles. This occurs when x=11.
Answer: B. x = 11
8. The figure will be a rhombus when all sides are congruent. The given figure has x=7, which will make all sides congruent.
Answer: D. x = 7
9. The given statement shows that opposite sides of a parallelogram are congruent.
Answer: A. Opposite sides of a parallelogram are congruent.
10. The figure will be a rectangle when all angles are right angles. To find the value of x for which this occurs, we need to equate the expressions for LN and MO and solve for x.
LN = MO
4x - 7 = 2x + 13
2x = 20
x = 10
Answer: A. x = 10
Quick Check Answers:
1. The figure is a rhombus because all sides are congruent.
Answer: B. Rhombus
2. For a parallelogram to be a rectangle, the diagonals must be congruent.
Answer: B. Diagonals are congruent
3. For a parallelogram to be a square, the diagonals must be perpendicular and congruent.
Answer: A. Diagonals are perpendicular ,and diagonals are congruent
4. To find the value of x for XYZA to be a rectangle, we need to find when all angles are right angles.
From the given figure, we can see that 4 is the value of x that satisfies this condition.
Answer: C. 4
I hope this helps! Let me know if you have any other questions.
To get the answers to these questions, you need to understand the conditions for rhombuses, rectangles, and squares. Here's an explanation of how to arrive at each answer:
Practice Answers:
1. To determine which statements are true, you need to know the properties of each shape. Statement 1 is false because a parallelogram is not always a rectangle. Statement 2 is true because a square is a special type of rhombus. Statement 3 is false because a rectangle is not always a square. Therefore, the correct answer is C. Statement 2 only.
2. Based on the given figure, you can determine the shape by analyzing its properties. Since all four sides of the figure are congruent, it is a rhombus. The correct answer is A. It is a rhombus.
3. To determine the value of x for which the figure is a rhombus, you need to apply the conditions for a rhombus. In a rhombus, all four sides are congruent. By setting the given expression equal to each other, you can solve for x. The correct answer is C. x = 12.
4. Similarly, to find the value of x for which the figure is a rectangle, you need to apply the conditions for a rectangle. In a rectangle, opposite sides are congruent. By setting the given expressions equal to each other, you can solve for x. The correct answer is A. x = 1.
5. To check if the bookshelf is rectangular, the carpenter can use the properties of a rectangle. She can measure the opposite sides of the bookshelf. If they are congruent, it indicates that the bookshelf is in the shape of a parallelogram. Then she can measure the diagonals. If they are congruent, it confirms that the bookshelf is rectangular. The correct answer is B. She can measure the opposite sides and diagonals.
6. If the diagonals of a parallelogram are both perpendicular and congruent, it satisfies the conditions for a square. Therefore, the correct answer is C. square.
7. By applying the conditions for a rectangle, which stipulate that opposite sides are congruent, you can set the given expressions equal to each other and solve for x. The correct answer is B. x = 11.
8. Similar to the previous question, by applying the conditions for a rhombus, which require all four sides to be congruent, you can set the given expressions equal to each other and solve for x. The correct answer is D. x = 7.
9. To complete the proof, you need to select the reason that justifies AB ≅ CF and BC ≅ AD. From the properties of a parallelogram, you know that opposite sides are congruent. Therefore, the correct answer is A. Opposite sides of a parallelogram are congruent.
10. To determine the value of x for which the figure is a rectangle, you need to apply the conditions for a rectangle, which state that opposite sides are congruent. Set the given expressions equal to each other and solve for x. The correct answer is A. x = 10.
Quick Check Answers:
1. To determine the most precise name for a figure, you need to identify its properties. From the given options, choose the name that best describes all the properties of the figure. In this case, a rhombus has equal sides and opposite angles, making it the most precise name. The correct answer is B. Rhombus.
2. For a parallelogram to be a rectangle, it must meet certain conditions. Among the given options, the one that states "Diagonals are congruent" is the correct answer. The diagonals of a rectangle are always congruent. Therefore, the correct answer is B. Diagonals are congruent.
3. Similarly, for a parallelogram to be a square, it must meet specific conditions. Among the given options, the one that states "Diagonals are perpendicular, and diagonals are congruent" is the correct answer. Both properties are necessary for a parallelogram to be a square. Therefore, the correct answer is A. Diagonals are perpendicular and diagonals are congruent.
4. To determine the value of x for which XYZA is a rectangle, you need to apply the conditions for a rectangle, which require opposite sides to be congruent. Set the given expressions equal to each other and solve for x. The correct answer is C. x = 4.
I hope this explanation helps clarify the process of arriving at the answers! Let me know if you have any further questions.