1. 8.5 Rhombi and Squares

2. Objectives  Recognize and apply properties of rhombi  Recognize and apply properties of squares

3. Rhombi  A rhombus is a parallelogram with four congruent sides.

4. Rhombi  Since rhombi are parallelograms, they have all the properties of a parallelogram.  In addition, they have 2 other properties which are theorems: - the diagonals of a rhombus are ┴ - each diagonal of a rhombus bisects a pair of opposite  s

5. Prove: D Given: BCDE is a rhombus, and Example 1:

6. Proof: Because opposite angles of a rhombus are congruent and the diagonals of a rhombus bisect each other, by the Reflexive Property and it is given that Therefore, by SAS. By substitution, Example 1:

7. Given: ACDF is a rhombus; Prove: Your Turn:

8. Proof: Since ACDF is a rhombus, diagonals bisect each other and are perpendicular to each other. Therefore, are both right angles. By definition of right angles, which means that by definition of congruent angles. It is given that so since alternate interior angles are congruent when parallel lines are cut by a transversal. by ASA. Your Turn:

9. Use rhombus LMNP to find the value of y if N Example 2a:

10. The diagonals of a rhombus are perpendicular. Substitution Add 54 to each side. Take the square root of each side. Answer: The value of y can be 12 or –12. Example 2a:

11. N Use rhombus LMNP to find if Example 2b:

12. Opposite angles are congruent. Substitution The diagonals of a rhombus bisect the angles. Answer: Example 2b:

13. Use rhombus ABCD and the given information to find the value of each variable. Answer: 8 or –8 Answer: a. b. Your Turn:

14. Squares  A square is a parallelogram with four congruent sides and four right angles.

15. Squares  If a quadrilateral is both a rhombus and a rectangle, then it is a square. parallelograms rhombi rectangles squares quadrilaterals

16. Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain. Explore Plot the vertices on a coordinate plane. Example 3:

17. Plan If the diagonals are perpendicular, then ABCD is either a rhombus or a square. The diagonals of a rectangle are congruent. If the diagonals are congruent and perpendicular, then ABCD is a square. Solve Use the Distance Formula to compare the lengths of the diagonals. Example 3:

18. Use slope to determine whether the diagonals are perpendicular. Since the slope of is the negative reciprocal of the slope of the diagonals are perpendicular.The lengths of and are the same so the diagonals are congruent. ABCD is a rhombus, a rectangle, and a square. Example 3:

19. ExamineThe diagonals are congruent and perpendicular so ABCD must be a square. You can verify that ABCD is a rhombus by finding AB, BC, CD, AD. Then see if two consecutive segments are perpendicular. Answer: ABCD is a rhombus, a rectangle, and a square. Example 3:

20. Determine whether parallelogram EFGH is a rhombus, a rectangle, or a square for E(0, –2), F(–3, 0), G(–1, 3), and H(2, 1). List all that apply. Explain. Your Turn:

21. Answer: and slope of slope of Since the slope of is the negative reciprocal of the slope of, the diagonals are perpendicular. The lengths of and are the same. Your Turn:

22. Let ABCD be the square formed by the legs of the table. Since a square is a parallelogram, the diagonals bisect each other. Since the umbrella stand is placed so that its hole lines up with the hole in the table, the center of the umbrella pole is at point E, the point where the diagonals intersect. Use the Pythagorean Theorem to find the length of a diagonal. A square table has four legs that are 2 feet apart. The table is placed over an umbrella stand so that the hole in the center of the table lines up with the hole in the stand. How far away from a leg is the center of the hole? Example 4:

23. The distance from the center of the pole to a leg is equal to the length of Example 4:

24. Answer: The center of the pole is about 1.4 feet from a leg of a table. Example 4:

25. Kayla has a garden whose length and width are each 25 feet. If she places a fountain exactly in the center of the garden, how far is the center of the fountain from one of the corners of the garden? Answer: about 17.7 feet Your Turn:

26. Assignment  Pre-AP Geometry  Pre-AP Geometry Pg. 434 #12 – 23, 26 – 31, 40 – 42  Geometry:  Geometry: Pg. 434 #12 – 23, 26 - 31