1. Warm Up(On Separate Sheet & Pass Back Papers)
Solve for x.
1. 16x – 3 = 12x + 13
2. 2x – 4 = 90
ABCD is a parallelogram. Find each measure.
3. CD 4. mC 4 47 14
104°

2. 6-4
Properties of Special Parallelograms
Holt Geometry

3. A rectangle is a quadrilateral with four right angles.

5. Example 1: Craft Application
A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.
Rect.  diags. 
KM = JL = 86
Def. of  segs.
 diags. bisect each other
Substitute and simplify.

6. A rhombus is a quadrilateral with four congruent sides.

8. Example 2B: Using Properties of Rhombuses to Find Measures
TVWX is a rhombus. Find a.
mVZT = 90°
Rhombus  diag. 
14a + 20 = 90°
Substitute 14a + 20 for mVTZ.
Subtract 20 from both sides and divide both sides by 14.
a = 5

9. A square is a quadrilateral with four right angles and four congruent sides.

10. Example 3: Verifying Properties of Squares
Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.

11. Example 3 Continued
Step 1 Show that EG and FH are congruent.
Since EG = FH,

12. Example 3 Continued
Step 2 Show that EG and FH are perpendicular.
Since ,

13. Example 3 Continued
Step 3 Show that EG and FH are bisect each other.
Since EG and FH have the same midpoint, they bisect each other.
The diagonals are congruent perpendicular bisectors of each other.

14. 6-5
Conditions for Special Parallelograms
Holt Geometry

16. Example 1: Carpentry Application
A manufacture builds a mold for a desktop so that , , and mABC = 90°. Why must ABCD be a rectangle?
Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since mABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem

18. 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.

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. Lesson Quiz: Part I
1. Given that AB = BC = CD = DA, what additional information is needed to conclude that ABCD is a square? 22

23. Lesson Quiz: Part I
A slab of concrete is poured with diagonal spacers. In rectangle CNRT, CN = 35 ft, and NT = 58 ft. Find each length.
1. TR CE
35 ft
29 ft

24. Lesson Quiz: Part II
PQRS is a rhombus. Find each measure.
3. QP mQRP 42
51°

25. Lesson Quiz: Part III
5. The vertices of square ABCD are A(1, 3), B(3, 2), C(4, 4), and D(2, 5). Show that its diagonals are congruent perpendicular bisectors of each other.

26. Lesson Quiz: Part IV
6. Given: ABCD is a rhombus.
Prove: 

27. Warm-Up
A slab of concrete is poured with diagonal spacers. In rectangle CNRT, CN = 35 ft, and NT = 58 ft. Find each length.
1. TR CE
35 ft
29 ft 27