warm up (on separate sheet & pass back papers) solve for x . 1. 16 x – 3 = 12 x + 13

Holt Geometry

6-4 Properties of Special Parallelograms

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. mC

4

47

14 104°

Holt Geometry



Holt Geometry

Holt Geometry


A rectangle is a quadrilateral with four right angles.

Holt Geometry


Holt Geometry


Example 1: Craft Application

A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.

Rect. diags.

Def. of segs.

Substitute and simplify.

KM = JL = 86

diags. bisect each other

Holt Geometry


A rhombus is a quadrilateral with four congruent sides.

Holt Geometry


Holt Geometry


Rhombus diag.

Example 2B: Using Properties of Rhombuses to Find Measures

TVWX is a rhombus. Find a.

Substitute 14a + 20 for mVTZ.

Subtract 20 from both sides and divide both sides by 14.

mVZT = 90°

14a + 20 = 90°

a = 5

Holt Geometry


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

Holt Geometry


Example 3: Verifying Properties of Squares

Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.

Holt Geometry


Example 3 Continued

Step 1 Show that EG and FH are congruent.

Since EG = FH,

Holt Geometry


Example 3 Continued

Step 2 Show that EG and FH are perpendicular.

Since ,

Holt Geometry


The diagonals are congruent perpendicular bisectors of each other.

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.

Holt Geometry


6-5 Conditions for Special Parallelograms

Holt Geometry

Holt Geometry


Holt Geometry


Example 1: Carpentry Application

A manufacture builds a mold for a desktop so that , , and mABC = 90°. Why must ABCD be a rectangle?

Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since mABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem 6-5-1.

Holt Geometry


Holt Geometry


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.

Holt Geometry


Example 3B: Identifying Special Parallelograms in the Coordinate Plane

W(0, 1), X(4, 2), Y(3, –2), Z(–1, –3)

Step 1 Graph WXYZ.

Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.

Holt Geometry


Step 2 Find WY and XZ to determine is WXYZ is a rectangle.

Thus WXYZ is not a square.

Example 3B Continued

Since , WXYZ is not a rectangle.

Holt Geometry


Step 3 Determine if WXYZ is a rhombus.

Example 3B Continued

Since (–1)(1) = –1, , PQRS is a rhombus.

Holt Geometry


Lesson Quiz: Part I

1. Given that AB = BC = CD = DA, what additional

information is needed to conclude that ABCD is a

square?

Holt Geometry


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 2. CE

35 ft 29 ft

Holt Geometry


Lesson Quiz: Part II

PQRS is a rhombus. Find each measure.

3. QP 4. mQRP

42 51°

Holt Geometry


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.

Holt Geometry


Lesson Quiz: Part IV

ABE CDF

6. Given: ABCD is a rhombus. Prove:

Holt Geometry


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 2. CE

35 ft 29 ft

warm up (on separate sheet & pass back papers) solve for x . 1. 16 x – 3 = 12 x + 13

Documents