Geometry unit 6.5
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1. The document provides examples and explanations for determining if a quadrilateral is a rectangle, rhombus, or square based on properties of its angles and sides. Key theorems relate special properties of parallelograms to specific shapes. 2. Examples demonstrate using the slope and length of diagonals to classify quadrilaterals in the coordinate plane as rectangles, rhombuses, or squares. Additional information like a quadrilateral being a parallelogram is needed to apply the theorems. 3. A lesson quiz tests understanding of classifying quadrilaterals and determining validity of conclusions based on given information.
Geometry unit 6.5
- 1. UUNNIITT 66..55 CCOONNDDIITTIIOONNSS FFOORR Holt Geometry RRHHOOMMBBUUSSEESS,, RREECCTTAANNGGLLEESS,, AANNDD SSQQUUAARREESS
- 2. Warm Up 1. Find AB for A (–3, 5) and B (1, 2). 2. Find the slope of JK for J(–4, 4) and K(3, –3). ABCD is a parallelogram. Justify each statement. 3. ÐABC @ ÐCDA opp. Ðs @ 4. ÐAEB @ ÐCED 5 –1 Vert. Ðs Thm.
- 3. Objective Prove that a given quadrilateral is a rectangle, rhombus, or square.
- 4. When you are given a parallelogram with certain properties, you can use the theorems below to determine whether the parallelogram is a rectangle.
- 5. Example 1: Carpentry Application A manufacture builds a mold for a desktop so that , , and mÐABC = 90°. Why must ABCD be a rectangle? Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since mÐABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem 6-5-1.
- 6. Check It Out! Example 1 A carpenter’s square can be used to test that an angle is a right angle. How could the contractor use a carpenter’s square to check that the frame is a rectangle? Both pairs of opp. sides of WXYZ are @, so WXYZ is a parallelogram. The contractor can use the carpenter’s square to see if one Ð of WXYZ is a right Ð. If one angle is a right Ð, then by Theorem 6-5-1 the frame is a rectangle.
- 7. Below are some conditions you can use to determine whether a parallelogram is a rhombus.
- 8. Caution In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram. To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43.
- 9. Remember! You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals.
- 10. Example 2A: Applying Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: Conclusion: EFGH is a rhombus. The conclusion is not valid. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. By Theorem 6-5-4, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. To apply either theorem, you must first know that ABCD is a parallelogram.
- 11. Example 2B: Applying Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: Conclusion: EFGH is a square. Step 1 Determine if EFGH is a parallelogram. Given EFGH is a parallelogram. Quad. with diags. bisecting each other
- 12. Example 2B Continued Step 2 Determine if EFGH is a rectangle. Given. EFGH is a rectangle. Step 3 Determine if EFGH is a rhombus. EFGH is a rhombus. with diags. @ rect. with one pair of cons. sides @ rhombus
- 13. Example 2B Continued Step 4 Determine is EFGH is a square. Since EFGH is a rectangle and a rhombus, it has four right angles and four congruent sides. So EFGH is a square by definition. The conclusion is valid.
- 14. Check It Out! Example 2 Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: ÐABC is a right angle. Conclusion: ABCD is a rectangle. The conclusion is not valid. By Theorem 6-5-1, if one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. To apply this theorem, you need to know that ABCD is a parallelogram .
- 15. Example 3A: Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)
- 16. Example 3A Continued Step 1 Graph PQRS.
- 17. Example 3A Continued Step 2 Find PR and QS to determine is PQRS is a rectangle. Since , the diagonals are congruent. PQRS is a rectangle.
- 18. Example 3A Continued Step 3 Determine if PQRS is a rhombus. Since , PQRS is a rhombus. Step 4 Determine if PQRS is a square. Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.
- 19. Example 3B: Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. W(0, 1), X(4, 2), Y(3, –2), Z(–1, –3) Step 1 Graph WXYZ.
- 20. Example 3B Continued Step 2 Find WY and XZ to determine is WXYZ is a rectangle. Since , WXYZ is not a rectangle. Thus WXYZ is not a square.
- 21. Example 3B Continued Step 3 Determine if WXYZ is a rhombus. Since (–1)(1) = –1, , PQRS is a rhombus.
- 22. Check It Out! Example 3A Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. K(–5, –1), L(–2, 4), M(3, 1), N(0, –4)
- 23. Check It Out! Example 3A Continued Step 1 Graph KLMN.
- 24. Check It Out! Example 3A Continued Step 2 Find KM and LN to determine is KLMN is a rectangle. Since , KMLN is a rectangle.
- 25. Check It Out! Example 3A Continued Step 3 Determine if KLMN is a rhombus. Since the product of the slopes is –1, the two lines are perpendicular. KLMN is a rhombus.
- 26. Check It Out! Example 3A Continued Step 4 Determine if PQRS is a square. Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.
- 27. Check It Out! Example 3B Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–4, 6) , Q(2, 5) , R(3, –1) , S(–3, 0)
- 28. Check It Out! Example 3B Continued Step 1 Graph PQRS.
- 29. Check It Out! Example 3B Continued Step 2 Find PR and QS to determine is PQRS is a rectangle. Since , PQRS is not a rectangle. Thus PQRS is not a square.
- 30. Check It Out! Example 3B Continued Step 3 Determine if KLMN is a rhombus. Since (–1)(1) = –1, are perpendicular and congruent. KLMN is a rhombus.
- 31. Lesson Quiz: Part I 1. Given that AB = BC = CD = DA, what additional information is needed to conclude that ABCD is a square?
- 32. Lesson Quiz: Part II 2. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: PQRS and PQNM are parallelograms. Conclusion: MNRS is a rhombus. valid
- 33. Lesson Quiz: Part III 3. Use the diagonals to determine whether a parallelogram with vertices A(2, 7), B(7, 9), C(5, 4), and D(0, 2) is a rectangle, rhombus, or square. Give all the names that apply. AC ≠ BD, so ABCD is not a rect. or a square. The slope of AC = –1, and the slope of BD = 1, so AC ^ BD. ABCD is a rhombus.
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