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Name: ____________________________________________ Block: __________ Date: _________ ID: A Honors Geometry Lesson 6-4 through 6-6 Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Classify the figure in as many ways as possible. A. B. C. D. rectangle, square, quadrilateral, parallelogram, rhombus rectangle, square, parallelogram rhombus, quadrilateral, square square, rectangle, quadrilateral ____ 2. Which statement is true? A. All squares are rectangles. B. All quadrilaterals are rectangles. C. All parallelograms are rectangles. D. All rectangles are squares. ____ 3. Lucinda wants to build a square sandbox, but she has no way of measuring angles. Explain how she can make sure that the sandbox is square by only measuring length. A. Arrange four equal-length sides so the diagonals bisect each other. B. Arrange four equal-length sides so the diagonals are equal lengths also. C. Make each diagonal the same length as four equal-length sides. D. Not possible; Lucinda has to be able to measure a right angle. ____ 4. In quadrilateral MNOP, ∠M ≅ ∠N. Which of a parallelogram, trapezoid, or rhombus could quadrilateral MNOP be? A. parallelogram or rhombus C. trapezoid only B. parallelogram only D. any of the three 1 Name: _______________________________________________________________________ ID: A Short Answer 5. In the rhombus, m∠1 = 15x, m∠2 = x + y, and m∠3 = 30z. Find the value of each variable. The diagram is not to scale. x=______ y=______ z=______ 6. Find the measure of the numbered angles in the rhombus. The diagram is not to scale. 7. ABCD is a rectangle. AC = 5x – 3 and BD = x + 5. Find the value of x and the length of each diagonal. x=______ AC=______ BD=______ 2 Name: _______________________________________________________________________ ID: A 8. In quadrilateral ABCD, m∠ACD = 2x + 4 and m∠ACB = 5x − 8. For what value of x is ABCD a rhombus? 9. In quadrilateral ABCD, AE = x + 6 and BE = 3x − 18. For what value of x is ABCD a rectangle? 10. ∠J and ∠M are base angles of isosceles trapezoid JKLM. If m∠J = 18x + 8, and m∠M = 11x + 15, find m∠K. 3 Name: _______________________________________________________________________ ID: A 11. The isosceles trapezoid is part of an isosceles triangle with a 42° vertex angle. a) What is the measure of an acute base angle of the trapezoid? b) Of an obtuse base angle? The diagram is not to scale. a)_____________ 12. LM is the midsegment of AB, LM and DC. x= ________ AB= ________ b)_____________ ABCD. AB = x + 8, LM = 4x + 3, and DC = 243. What is the value of x? Find LM= ________ 4 DC= ________ Name: _______________________________________________________________________ 13. Find the measures of the numbered angles in the kite. The diagram is not to scale. _____________ _____________ _____________ 14. Find the values of the variables and the lengths of the sides of this kite. x=______ y=______ AB=______ BC=______ CD=______ 5 DA=______ ID: A Name: _______________________________________________________________________ ID: A 15. Find the values of a and b. The diagram is not to scale. 16. What is the most precise name for quadrilateral ABCD with vertices A(–5, 2), B(–3, 5), C(4, 5), and D(2, 2)? 6 Name: _______________________________________________________________________ ID: A Essay 17. Explain how you can determine, without measuring any angles, whether a quadrilateral is a rectangle. Other 18. Two consecutive angles of a quadrilateral are right angles, but the quadrilateral is not a rectangle. Can the quadrilateral be a parallelogram? Explain. 19. ABCD is a rhombus. Write a statement reason proof to show that ΔABC ≅ ΔCDA. 7 ID: A Honors Geometry Lesson 6-4 through 6-6 Answer Section MULTIPLE CHOICE 1. 2. 3. 4. ANS: ANS: ANS: ANS: A A B D TOP: TOP: TOP: TOP: 6-4 6-4 6-5 6-6 Problem 1 Classifying Special Parallelograms Problem 1 Classifying Special Parallelograms Problem 3 Using Properties of Parallelograms Problem 1 Finding Angle Measures in Trapezoids SHORT ANSWER 5. ANS: x = 6, y = 84, z = 3 TOP: 6-4 Problem 2 Finding Angle Measures 6. ANS: m∠1 = 90, m∠2 = 24, and m∠3 = 66 TOP: 6-4 Problem 2 Finding Angle Measures 7. ANS: x = 2, DF = 7, EG = 7 TOP: 6-4 Problem 3 Finding Diagonal Length 8. ANS: 4 TOP: 6-5 Problem 2 Using Properties of Special Parallelograms 9. ANS: 12 TOP: 6-5 Problem 2 Using Properties of Special Parallelograms 10. ANS: 154 TOP: 6-6 Problem 2 Finding Angle Measures in Isosceles Trapezoids 11. ANS: 69°; 111° TOP: 6-6 Problem 2 Finding Angle Measures in Isosceles Trapezoids 12. ANS: 35 TOP: 6-6 Problem 3 Using the Midsegment of a Trapezoid 1 ID: A 13. ANS: m∠1 = 17, m∠3 = 73 TOP: 6-6 Problem 4 Finding Angle Measures in Kites 14. ANS: x = 9, y = 14; 11, 20 TOP: 6-6 Problem 4 Finding Angle Measures in Kites 15. ANS: a = 115, b = 71 TOP: 6-6 Problem 1 Finding Angle Measures in Trapezoids 16. ANS: parallelogram TOP: 6-4 Problem 1 Classifying Special Parallelograms ESSAY 17. ANS: [4] Shows enough properties that do not require angle measurement and concludes, based on those properties, that the quadrilateral is a rectangle. Sample: Measure to show diagonals bisect each other. This makes the quadrilateral a parallelogram. Measure to show that diagonals are congruent. This makes the parallelogram a rectangle. [3] demonstrates understanding of exercise, but omits one property needed to conclude quadrilateral is parallelogram [2] gives way of determining that quadrilateral is rectangle, but includes angle measurement (such as making right angles) [1] gives only one step necessary for concluding quadrilateral is rectangle TOP: 6-5 Problem 3 Using Properties of Parallelograms OTHER 18. ANS: No; if it were a parallelogram, then the fact that it has two consecutive right angles would mean that it has four right angles and would have to be a rectangle. TOP: 6-5 Problem 1 Identifying Special Parallelograms 19. ANS: AB ≅ CD and BC ≅ DA by the definition of rhombus. AC ≅ AC by the Reflexive Property, so ΔABC ≅ ΔCDA by SSS. TOP: 6-5 Problem 2 Using Properties of Special Parallelograms 2